In this talk I will provide a short introduction to a class of operator splitting methods with complex coefficients which possess a special symmetry, the so-called symmetric-conjugate methods, and analyze their application for the time integration of linear evolution problems. Including complex coefficients with non-negative real parts permits the design of favorable high-order schemes that remain stable in the context of parabolic problems. This sets aside the second-order barrier for standard splitting methods with real coefficients as well as the fourth-order barrier for modified splitting methods involving double commutators. Relevant applications include nonreversible systems and ground state computations for Schr{\"o}dinger equations based on the imaginary time propagation method.

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Complex operator splitting methods for Ginzburg-Landau equations and related problems" (2024)

The evolution of most quantum systems is modeled by differential equation in the complex space. However, in general, the equations are numerically solved using integrators with real coefficients. To consider complex coefficients usually does not make the schemes computationally more costly and can provide more accurate results. In this talk, we explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. There are pros (high accuracy and not to increase the cost) and cons (instability and loose of qualitative properties) when using complex coefficients. However, there is a class of methods with complex coefficients with a particular symmetry that keep most pros while avoid most cons. This class of integrators are stable and are conjugate to unitary methods for sufficiently small step sizes. These are promising methods that we will explore: we build new methods and we analyse their performance on several examples. This is joint work with Joakim Bernier, Fernando Casas and Alejandro Escorihuela.

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Complex operator splitting methods for Ginzburg-Landau equations and related problems" (2024)

In this presentation, three projects are sketched: 1) F-Plane Approximation for Solar Simulation: We introduced the f-plane approximation into the ANTARES code and ran 3-D solar simulation with and without rotation, attempting to analyze how much influence rotation would have on the convection structures. 2) Linear Stability Analysis of an Incompressible Fluid with Vertical Shear in F-plane: Shear instability and GSF instability are two possible sources of extra mixing in stars to explain the differences between theoretical stellar evolution models and observations. We consider an incompressible fluid with vertical shear in f-plane and try to find out what really happens when shear instabilities coexist with GSF instabilities. 3) The Stability Analysis of Polar Cyclones on Saturn and Jupiter: The long-lived cyclones in the polar regions of Jupiter and Saturn have been explored for a long time. We aim to explain the number and location of Jupiter’s circumpolar cyclones and the absence of those on Saturn by linear stability analysis. This is ongoing joint work with O. Koch, F. Kupka and N.J. Mauser.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Working group on 'Numerical methods for solar simulations' (2024)

We consider equations of motion associated to a model in astrophysics. The PDE are firstly semi-discretized in space and subsequently integrated in time by IMplicit-EXplicit (IMEX) Runge-Kutta methods constructed to preserve different stability properties. For some simple examples, as well as for the problem of double-diffusive convection, it can be demonstrated that they provide a significant computational advantage over other methods from the literature. Ongoing work continues with the study of implementation issues as well as with the study and construction of robust IMEX schemes for some other problems in astrophysics. This is joint work with O. Koch and F. Kupka.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Working group on 'Numerical methods for solar simulations' (2024)

We discuss some aspects of the effort to produce a "digital twin" of the earth climate. The status of ultra-high-resolution numerical climate modelling and recent computational achievements are discussed. Mathematical challenges and opportunities arise when numerical models aim to represent an increasing number of turbulent scales. These challenges comprise the PDE of atmosphere and ocean dynamics, their numerical discretization and the modelling of (still) unresolved scales. This is part of the research of the DFG Forschungsgruppe FOR 5528: Mathematical Study of Geophysical Flow Models: Analysis and Computation.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: (Generalized) Hydrodynamics and kinetic equations: (numerical) modeling (2023)

I take the audience on a scientific journey from the physics of neutron stars to cosmology and further on to turbulence and its role for oceaongraphy and climate modeling. Scientific highlights on this journey include an exact equation of state for neutron stars, results on cosmology, and a general turbulence model which has guided the modeling of transport processes in oceanography which is needed in climatology. From short encounters to longterm collaborations famous physicists are part of this story, including P.A.M. Dirac, W. Heisenberg, I. Rabi, J.A. Wheeler and many others.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

The atmospheres of most stars have at least some level of magnetic activity. This is modulated by variability, which manifests itself as varying magnetic strength across the stellar surface and in time as well as in the form of different magnetic behaviour of different stars. This is moreover intertwined with all the other physical effects occurring in the atmospheres of stars, in particular convection, radiative transfer and turbulence. In the case of the Sun, magnetic fields are known to be ubiquitous, at an average level of roughly 1 hG across its surface, which - inter alia - has an impact on its inferred temperature stratification and chemical abundance. It is especially interesting to understand solar magnetism, for example its main magnetic cycle, also in comparison to other stars, given the Sun's driving influence on life on Earth and as the base energy input for terrestrial climate. Knowledge of stellar activity is also crucial for improved exoplanet detection and characterisation. Our team is focussing on different aspects of stellar atmosphere physics, from the viewpoint of numerical (magneto-)hydrodynamic simulations. Recent examples include the production of models for stars of spectral type F to A, and the study of hard turbulence as possible driver of synchronised swaying atmospheric motions akin to the still unexplained effect of solar supergranulation.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

The overall understanding of solar and stellar convection has been questioned during the last decade or so with helioseismic results suggesting that the convective amplitudes at large horizontal scales in the Sun might be much lower than indicated by current simulations or mixing length estimates. A manifestation of this ``convective conundrum'' is that global simulations struggle to reproduce a solar-like differential rotation with a fast equator and slow poles with nominally solar parameters. A major contributor to this is that giant cell convection, with characteristic length scale comparable to the depth of the convection zone, is excited in simulations but appears to be much weaker in the Sun. A possible solution to this conundrum is that a large fraction of the solar convection zone is in fact stably stratified due to plumes originating near the surface and piercing the whole convection zone, such that giant cells are not excited. Non-rotating numerical simulations lend support to such non-local scenario of convection and lead to sizeable Schwarzschild-stable, yet convecting, layers in deep convection zones. Another possibility is that convection is rotationally constrained such that horizontal extent of convection cells is significantly reduced. New results from hydrodynamic rotating Cartesian convection simulations are presented that seek to capture the rotationally constrained convection near the base of the solar convection zone. The current results indicate that in models corresponding to the deep parts of the solar convection zone, the subadiabatic and overshoot layers are somewhat shallower than in the non-rotating case. Furthermore, these simulations suggest that deep convection in the Sun is not strongly rotationally constrained and that rotational suppression of large scale flows is weak.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

In 1D stellar evolution models, the process of convection is often described using the mixing length theory (MLT). However, MLT does not account for the non-locality of convection, and an ad hoc implementation of overshooting is needed. The Kuhfuss theory is one of the theories that attempts to derive a more complete picture of turbulent convection. In this theory, non-locality is not implemented artificially, but is included in the theory. Both versions of the Kuhfuss theory, the 1-equation model as well as the 3-equation model, are implemented in the stellar evolution code GARSTEC and have already been tested on convective cores on the main sequence before (Ahlborn et al. 2022). Following these promising results for convection in stellar cores, we tested the Kuhfuss theory for convective envelopes. We applied the 1-equation model of the Kuhfuss convection theory to a stellar model calibrated to the Sun. Using helioseismic measurements of quantities of the convective envelope and interior structure, we quantified the differences and improvements from the Kuhfuss theory compared to MLT. We furthermore followed the evolution of stars to the red giant branch to study the influence of the Kuhfuss theory on the location of the red giant branch bump, which is known to be sensitive to the description and depth of convective overshooting. In the future, these cases will also be studied using the full 3-equation Kuhfuss model.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

Observations of stars with convectively burning cores have shown that the size of these cores is substantially underestimated. The increase of the convective core size, known as overshooting, has profound effects on the stellar structure and evolution, e.g. affecting age estimates, luminosities or nucleosynthetic yields of stars. Here, we applied a turbulent convection theory to model the evolution of intermediate and high-mass stars. We predict the emergence of an overshooting zone and modifications to the thermal stratification. The application of a turbulent convection theory is a crucial step towards a more realistic description of convection in stellar models. The predictions of the turbulent convection model may be tested against a variety of different observations, e.g. spectroscopic observations of massive stars, asteroseismic observations or observations of detached binary systems. Finally, the predictions of the turbulent convection model can be compared to hydrodynamic simulations of turbulent convection.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

In this presentation I shall provide an overview of our current understanding of modelling energy exchange between stellar convection and oscillations in stars supporting solar-type oscillations. Stellar calculations, adopting a 1D, non-local, time-dependent convection model, are calibrated against seismic observations and 3D-simulation results. These stellar models are tested against data from the Sun and from Kepler main-sequence stars. This provides insight into the physical processes that determine energy transport in the outer stellar layers and to a better understanding of the so-called surface effects of pulsation frequencies.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

While all of the stars change their brightness during their lifetimes, there are many among them that do this on a human timescale (from less than a day to years) due to external or internal reasons. We call those stars classical variables, which exhibit a strong, stable radial pulsation with periods from 0.3-100 days. In these cases, the outer envelope of the star periodically expands and shrinks due to an effect tied to hydrogen and helium ionisations called the kappa mechanism. They are important to astronomers because their period is proportional to their average brightness, making them perfect distance indicators. Since the first electronic computers became available, astronomers have applied them to model the structure and dynamics of these (and every other) types of stars. The first attempts neglected convection and turbulence, considering only radiative energy transport. However, it soon turned out that we could not adequately describe pulsation without convection. Moreover, the different improved forms of static mixing length theory were also inadequate. Hence, massive research was started to create a time-dependent theory that can describe convection correctly in a one-dimensional approximation. These efforts revealed some hidden features of the phenomena but could not answer all of the questions raised. Since convection and non-radial pulsation are genuinely multi-dimensional phenomena, multi-D models seem inevitable, but this approach requires high computational performance, which was not available decades ago. Today, though we have better equipment, numerical modelling of turbulent convection in stars is still a great challenge due to the many magnitudes of scale it involves, especially in classical pulsators. In this talk, I highlight some of the achievements of this journey and show the recent developments and future aspects of turbulent RHD modelling in classical pulsating stars.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

We speak about numerical issues and results regarding the simulation of solar granulation flows and the pulsation-convection interaction in Cepheids in 3D and 2D, respectively.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

In intention of this talk is to show how research on a rather specific topics from stellar astrophysics, the study of atmospheres of A-type stars, has led myself to numerous collaborations with researchers working in other fields such as meteorology, oceanography, numerical mathematics, and high preformance computing. To explain "turbulence" in the context of solar and stellar astrophysics, a short introduction into simulations of solar granulation will be given (much more details will follow in Herbert Muthsam's talk) followed by how turbulent convection is detected and modelled in A-type stars. Various modelling approaches have been used in this context: mixing lenth theory, two-point closure models, Reynolds stress models, and numerical simulations. The latter lead to the necessity to develop improved time integration methods which have first been probed in studies of semi-convection (diffusive convection). Studies in meteorology inspired new models for higher order moments required for Reynolds stress models. Finally, some result on the modelling of convective overshooting is presented which has been inspired by work that will be discussed in detail in other talks during the workshop (by Felix Ahlborn, Teresa Braun, Petri Käpylä).

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Workshop: Modelling of geophysical and stellar flows (2023)

Three examples from geophysical fluid dynamics will showcase mathematical modelling as the "art of judicious simplification": The computational prediction of two seasonal to decadal phenomona, the "quasi-biennial oscillation" (QBO) and the "El Niño Southern Oscillation" (ENSO) became possible only after theoreticians had captured their essential causal structures in convincing reduced mathematical models. With our own research, we aim to similarly untangle the mechanisms behind the "rapid intensification" (RI) of tropical storms during their transition to hurricane strength.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

Turbulence closure models (parameterization schemes) currently used in numerical models of the atmosphere are discussed. The focus is on truncated one-equation turbulence kinetic energy (TKE) closure schemes that are arguably the present-day draft horses of operational meteorology, e.g., numerical weather prediction. Advantages and shortcomings of one-equation TKE schemes are outlined in the context of various operational constraints. A TKE scalar variance (TKESV) closure scheme is considered in some detail. The TKESV scheme carries transport equations (with due regard for the time-rate-of-change and third-order transport terms) for both the TKE and the variances and covariance of scalar quantities (e.g., temperature and humidity) that characterize turbulence potential energy. It is argued that the TKESV scheme has considerable advantages over the TKE scheme in terms of the essential physics but it can still meet severe operational requirements. Careful consideration is given to a number of tricky parameterization issues, including the pressure-scrambling effects in the Reynolds-stress and scalar-flux equations and the influence of clouds on turbulent mixing. An assumed PDF (probability distribution function) closure approach is briefly outlined. Finally, realizability of turbulence closures is considered within a more general framework of the problem of moments of the probability theory.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

The three principal dynamical regimes of the atmosphere and the ocean are: i) small-scale turbulence down to the smallest space and time scales ii) internal gravity waves over a wide range of spatial scales iii) geostrophically balanced eddying motion at the largest space and time scales. All regimes are of turbulent character and need parameterisations in ocean components of climate models because of the lack of coarse grid resolution. A few aspects of closures for gravity wave turbulence are presented and closures for eddies in the ocean are discussed.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

Stellar evolution models provide the foundation of several methods applied to study the evolutionary properties of stars and stellar populations, both Galactic and extragalactic. The accuracy of the results obtained with these techniques is tied to the accuracy of the stellar models, and in this context the correct treatment of turbulent convection is crucial. Unfortunately, the modelling of turbulent convection in stellar evolution computations is still affected by sizable uncertainties. The aim of this talk is to highlight the effect of turbulent convection on the most important stellar model predictions in the context of the study of stellar systems like star clusters and galaxies, and the (simple) prescriptions we currently use (out of necessity).

• Thematic program: Mathematics – Magnetism - Materials (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

In a viscous fluid, the energy dissipation is the signature of the breaking of the time-reversal symmetry (hereafter TSB) t->-t, u-> -u, where u is the velocity. This symmetry of the Navier-Stokes equations is explicitly broken by viscosity. Yet, in the limit of large Reynolds numbers, when flow becomes turbulent, the non-dimensional energy dissipation per unit mass becomes independent of the viscosity, meaning that the time-reversal symmetry is spontaneously broken. Natural open questions related to such observation are: what is the mechanism of this spontaneous symmetry breaking? Can we associate the resulting time irreversibility to dynamical processes occurring in the flow? Can we devise tools to locally measure this time irreversibility? In this talk, I first show that the TSB is indeed akin to a spontaneous phase transition in the Reversible Navier-Stokes equations, a modification of the Navier-Stokes equation suggested by G. Gallavotti to ensure energy conservation and relevance of statistical physics interpretation. I then discuss a mechanism of the TSB in Navier-Stokes was first suggested by L. Onsager in 1949, in which quasi-singularities or singularities create a non-viscous dissipation. I exhibit the tools to track these quasi-singularities. I show how the application of these tools to velocity measurements in a turbulent swirling flow allows to detect Eulerian and Lagrangian signatures of irreversibility. This enables me to evidence the structures that are responsible for irreversibility and associate them with peculiar properties of the local velocity field or trajectories.

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: Pauli Symposium on Turbulence in Stellar & Geophysical flows (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Numercial models in Biology and Medicine (2023/2024)
• Event: Workshop on "Mathematical Models in Medicine" (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2023/2024)
• Event: 14th Plasma Kinetics Working Group Meeting (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

• Thematic program: Quantum Mechanics (2023/2024)
• Event: Workshop on "Tensor Network Methods in Quantum Dynamics“ (2023)

We introduce a stochastic interacting particle consensus system for global optimization of high dimensional non-convex functions. This algorithm does not use gradient of the function thus is suitable for non-smooth functions. We prove, for fully discrete systems, that under dimension-independent conditions on the parameters, with suitable initial data, the algorithms converge to the neighborhood of the global minimum almost surely. We also introduce an Adaptive Moment Estimation (ADAM) based version to significantly improve its performance in high-space dimension.

• Thematic program: (Physics informed) Machine Learning and Uncertainty Quantification (2022/2023)

In this talk, I’ll introduce a simple new way – called „Schrödingerisation“ – to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the „warped phase transformation“, any linear partial differential equation can be recast into a system of Schrödinger’s equations – in real time — in a straightforward way. This can be seen directly on the level of the dynamical equations without more sophisticated methods. This approach is not only applicable to PDEs for classical problems but also those for quantum problems – like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit.

• Thematic program: (Quantum) Wave equations (2022/2023)

Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper: 1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly. 2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent math- ematical research offers advanced data analysis techniques that could be “game changing” in this respect. 3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical for- malization can help structuring discussions and bridging language barriers in interdisciplinary research.

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: 22. Pauli Colloquium on "Mathematics for Climate research":; Rupert Klein (FU Berlin) (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

• Thematic program: (Quantum) Wave equations (2022/2023)
• Event: Workshop on "Nonlinear dispersive equations - Inverse scattering and PDE methods" (2022)

I first report on recent numerical experiments with time-dependent tree tensor network algorithms for the approximation of quantum spin systems. I will then describe the basics in the design of time integration methods that are robust to the usual presence of small singular values, that have good structure-preserving properties (norm, energy conservation or dissipation), and that allow for rank (= bond dimension) adaptivity and also have some parallelism. This discussion of basic concepts will be done for the smallest possible type of tensor network differential equations, namely low-rank matrix differential equations. Once this simplest case is understood, there is a systematic path to the extension of the integrators and their favourable properties to general tree tensor networks. This talk is based on joint work with many colleagues and former and present students, among which I wish to single out Othmar Koch for the first mathematical work on dynamical low-rank approximation (DLRA) in 2007, Ivan Oseledets for jointly finding the first robust DLRA integrator (the projector-splitting integrator) in 2014, Gianluca Ceruti for jointly finding the Basis Update & Galerkin (BUG) integrators in 2021, and him and Hanna Walach and Dominik Sulz for the recent systematic extension from low-rank matrices to general tree tensor networks.

• Thematic program: (Quantum) Wave equations (2022/2023)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

• Thematic program: Models in Plasmas, Earth and Space Science (2022/2023)
• Event: Workshop on "Seamless numerics for geophysical flow models" (2022)

The aim of quasilinear theory is to explain relaxation or saturation of kinetic instabilities governed by the Vlasov-Poisson (VP) equation, by showing that in fact the Hamiltonian dynamics of VP can be approximated by a diffusion equation in velocity for the space-average distribution function.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

Based on a series of works in collaboration with Gilles Lebeau and Nicolas Burq -Propagation of smallness and control for heat equations (with Nicolas Burq, to appear in JEMS), -Spectral Inequalities for the Schrödinger operator (with Gilles Lebeau). -Propagation of smallness and spectral estimates (with Nicolas Burq) And the recent advances in propagation of smallness introduced by Logonuv and Malinnikova. A. Logunov and E. Malinnikova. Quantitative propagation of smallness for solutions of elliptic equations. Preprint, Arxiv, (arXiv:1711.10076), 2017 A. Logunov. Nodal sets of Laplace eigenfunctions : polynomial upper estimates of the Hausdorff measure. Ann. of Math. (2), 187(1):221–239, 2018.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

The Pauli-Poisson equation is a semi-relativistic description of electrons under the influence of a given linear (strong) magnetic field and a self-consistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schrödinger type equations for the two components of the spinor, coupled by the additional Stern-Gerlach term of magnetic field and spin represented by the Pauli matrices. On the other hand the Pauli-Poiswell equation includes the self-consistent description of the magnetic field by coupling it via a three Poisson equations with the Pauli current as source term to the Pauli equation. The Pauli-Poiswell equation offers a fully self-consistent description of spin-1/2-particles in the semi-relativistic regime. We introduce the equations and study the semiclassical limit of Pauli-Poisson towards a semi-relativistic Vlasov equation with Lorentz force coupled to the Poisson equation. We use Wigner transform methods and a mixed state formulation, extending the work of Lions-Paul and Markowich-Mauser on the semiclassical limit of the Schrödinger-Poisson equation. We also present a result on global weak solutions of the Pauli-Poiswell equation.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

This talk presents a joint mean-field and classical limit by which the Euler-Poisson system is rigorously derived from the N-body Schrödinger equation with Coulomb interaction in space dimension 3. One of the key ingredients in this derivation is a remarkable inequality for the Coulomb potential which has been obtained by S. Serfaty in 2020 (Duke Math. J.). 2)

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in Hs, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold s = 1/6. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity. This talk is based on a joint work with Razvan Mosincat (University of Bergen).

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

The majority of dispersive equations in one space-dimension can be realized as dispersive perturbations of the Burgers equation ut + uux = Lux, where L is a local or nonlocal symmetric operator. For negative order dispersion, the Burg- ers’ nonlinearity dominates and classical solutions break down due to shock-formation/wave- breaking. Using hyperbolic techniques we establish global existence and uniqueness of entropy solutions, with L2 initial data, for a family of negative order dispersive equations, but our main focus will be on a new generalization of the classical Oleïnik estimate for Burgers equation. We obtain one sided Hölder regularity for the solutions, which in turn controls their height and provides a novel bound of the lifespan of classical solutions based on their initial skewness. This is joint work with Jun Xue (NTNU).

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blow-up. A finite-time blow-up of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We are concerned with finding Fokker-Planck equations in whole space with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrary close to its infimum. This infimum is 1, corresponding to the high-rotational limit in the Fokker-Planck drift. Such an optimal Fokker-planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the L2- projector norms of the Fokker-Planck equation and of its drift-ODE coincide. Finally we give an outlook onto using Fokker-Planck equation with t-dependent coefficients. This talk is based on a joint work with Beatrice Signorello.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

Currently, the studies on periodic travelling waves of the nonlinear dispersive equations are becoming very popular. In this study we investigate the spectral stability of the periodic waves for the fractional Benjamin-Bona-Mahony (fBBM) equation, numerically. For the numerical generation of periodic travelling wave solutions we use an iteration method which is based on a modification of Petviashvili algorithm. This is a joint work with S. Amaral, H. Borluk, G.M. Muslu and F. Natali.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We present a detailed numerical study of solutions to Davey-Stewartson (DS) II systems, nonlocal non-linear Schrödinger equations in two spatial dimensions. A possible blow-up of solutions is studied, a conjecture for a self-similar blow-up is formulated. In the integrable cases, numerical and hybrid approaches for the inverse scattering are presented.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

Using the Birkhoff map of the Benjamin-Ono equation as a starting point, we deform it near an arbitrary compact family of finite dimensional tori, invariant under the Benjamin-Ono flow, so that the following main properties hold: (i) When restricted to the family of finite dimensional tori, the transformation coincides with the Birkhoff map. (ii) Up to a remainder term, which is smoothing to any given order, it is a pseudo-differential operator of order 0, with principal part given by the Fourier transform, modified by a phase factor. (iii) The transformation is canonical and the pullback of the Benjamin-Ono Hamiltonian by it is in normal form up to order three. Such coordinates are a key ingredient for studying the stability of finite gap solutions of arbitrary size of the Benjamin-Ono equation under small, quasi-linear, momentum preserving perturbations. This is joint work with Riccardo Montalto.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

For solutions of the Benjamin-Ono equation with periodic boundary conditions, I will discuss the link in the high frequency regime between the nonlinear Fourier transform inherited from the integrable structure, and a gauge transform introduced by T. Tao in 2004 in the context of the low regularity initial value problem. As an application, we will get optimal high frequency approximations of solutions. This talk is based on a recent joint work with T. Kappeler and P. Topalov.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We survey new and old results on the Intermediate Long Wave (ILW) equation from modeling, PDE and integrability aspects.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

The talk deals with the restricted Grassmannian which is a Hilbert manifold and related Banach Lie-Poisson spaces. One of the integrable systems related to this setup is of course the KdV equation. Using Magri method it is also possible to define another infinite hierarchy of differential equations on a certain central extension of a Banach Lie-Poisson space. Using integral of motions it is possible to write down solutions in particular cases.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We define the notion of Morita equivalence for singular Riemannian foliations (SRFs) such that the underlying singular foliations are Hausdorff-Morita equivalent as recently introduced by Garmendia and Zambon. We then define a functor from SRFs to the category of I-Poisson manifolds, where the objects are Poisson manifolds together with appropriate ideals and morphisms are defined as a particular relaxation of Poisson maps. We show that Morita equivalent SRFs are mapped to I-Poisson manifolds with isomorphic Poisson algebra of smooth functions on the symplectically reduced spaces. This is joint work in progress with T. Strobl.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Exponential maps arise naturally in Lie theory and in the context of smooth manifolds endowed with affine connections. The Poincaré--Birkhoff--Witt isomorphism and the complete symbols of differential operators are related to these classical exponential maps through their infinite-order jets. The construction of (jets of) exponential maps can be extended to differential graded (dg) manifolds. As a consequence, the space of vector fields of any dg manifold inherits an L-infinity algebra structure, which is related to the Atiyah class of the dg manifold. Specializing this construction to the dg manifold arising from a foliation of a smooth manifold, one obtains an L-infinity structure on the de Rham complex of the foliation. In particular, a complex manifold can be regarded as a sort of `complexified' foliation. It turns out that the induced L-infinity structure is quasi-isomorphic to the L-infinity structure associated to the Atiyah class of the holomorphic tangent bundle on the Dolbeault complex first discovered by Kapranov. This is a joint work with Mathieu Stiénon and Ping Xu.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk I will discuss how one can construct an infinite dimensional space of paths associated to a sheaf of Lie-Rinehart algebras. We will briefly examine some of the topological properties of this path space and how it can be used to construct a diffeological groupoid which appears to integrate the underlying sheaf. We will also take a look at some motivating examples for studying sheaves of Lie-Rinehart algebras over manifolds.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk I will present the diffeological space of paths along a singular foliation and its groupoid structure. I will also show how to construct the fundamental groupoid of a singular foliation from its diffeological space of paths. This is a presentation of the joint work with Joel Villatoro entitled "Integration of singular foliations via paths" and to be published on IMRN.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

I will discuss different aspects of infinite-dimensional symplectic geometry. Why is it interesting and what are important applications? What are the common technical issues in the infinite-dimensional setting and how to overcome them? In particular, I will explain how the Marle-Guillemin-Sternberg local normal form and symplectic reduction work in infinite dimensions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Quantizations of infinitesimal gauge symmetries are classified in terms of the continuous Lie algebra cohomology group of gauge algebras in degree 2. For gauge bundles with semisimple fibers, this space was calculated by Janssens-Wockel (2013), their method relying heavily on the low degree of the cohomology group. In this talk, we extend these results to homology in higher degree. To this end, we review some homological algebra for topological chain complexes and use it to lift the well-known Loday-Quillen-Tsygan-Theorem (1983, 1984) from a statement in algebraic Lie algebra homology to one that takes topological data into account. For globally trivial gauge algebras whose fibres are classical Lie algebras, this calculates a certain stable part of continuous homology. A similar description was given by Feigin (1988), but lacking a detailed proof. Finally, we use the results for trivial bundles to construct a Gelfand Fuks-like local-to-global spectral sequence from which homological information about nontrivial gauge algebras can be extracted. If time permits, we discuss obstructions to a full understanding of this spectral sequence. This talk is based on joint work with Bas Janssens.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Analytic structure on a manifold (adapted to a specific analytic atlas) is a special type of G-structure of infinite order. I will report on work in progress that aims to answer the following questions: What is an almost analytic structure? What are obstructions to integrability? Does formal integrability imply integrability? What natural geometric objects define corresponding analytic structures?

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external homepage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Local normal form theorems in differential geometry are often the manifestation of rigidity of the structure in normal form. For example, the existence of local Darboux coordinates in symplectic geometry follows from the fact that, locally, the standard symplectic structure has no deformations. After introducing closed pseudogroups and their associated sheaf of Lie algebras, I will discuss a general local rigidity result for solutions to PDE’s under the action of a closed pseudogroup of symmetries. The result is of the form: “infinitesimal tame rigidity” implies “tame rigidity”; it is in the smooth setting, and the proof uses the Nash-Moser fast convergence method. Several classical theorems fit in our setting: e.g. the Newlander-Nirenberg theorem in complex geometry, Conn’s theorem in Poisson geometry. This is a joint work with Roy Wang.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk we outline how one can use the Nash-Moser method to prove Poisson linearization results of compact semisimple Lie algebras. We use Conn's idea to prove a more general linearization result.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

A Jacobi structure is a Lie bracket on the sections of a line bundle. These brackets encode time-dependent mechanics in the same way Poisson brackets encode mechanics. Contact groupoids are finite-dimensional models for the "integrations" of these infinite-dimensional Lie algebras. In this talk, we explain how, under a certain compactness hypothesis, one can adapt the argument of Gray-Moser to these multiplicative contact structures and point out some applications.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Infinite-dimensional differential geometry is often viewed as a fairly arcane subject with little connection to geometric questions arising in (finite-dimensional) applications. The aim of this talk is to show that this impression could not be further from the truth. We will take a scenic tour to a multitude of examples, connecting finite, infinite-dimensional and higher geometry. While some of these are well known classics such as Euler-Arnold theory for partial differential equations, also new results with surprising applications (such as in rough path integration theory) will be presented. As this talk is intended as a gentle introduction to these topics, no prior knowledge of infinite-dimensional geometry will be necessary.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

The evolution of cold dark matter (CDM) is governed by the cosmological Vlasov–Poisson equations. As it is well-known, the gravitational collapse of CDM leads to infinite-density caustics that seed the primordial dark-matter halos in the cosmic large-scale structure. Focusing on the one-dimensional case, I report a landscape of so far unknown singularities in the particle acceleration that emerge after the first crossing of particle trajectories. These singular features may be regulated by assuming a finite temperature for dark matter, which, to some extend, simplifies the numerical computation but complicates the theoretical modelling. Alternatively, singular features are naturally tamed in semiclassical, Schrödinger-like descriptions for the large-scale structure which I will discuss as well.

• Thematic program: Models in Plasmas, Earth and Space Science (2019/2020)
• Event: "Experimental and Numerical Quantum Cosmology". (2020)

Designing effective field theories in a laboratory setup has gained increasing attention over the last years and lies at the heart of analog-gravity experiments. Probing the validity of these effective models constitutes an essential step towards (quantum) simulators of otherwise inaccessible systems. Focusing on its applications for early universe cosmological problems, I report on the opportunities, validation, and limitations of analog classical and quantum simulators for (Quantum) Field Theory in curved and time-dependent spacetimes, in particular to cosmic inflation. As specific examples, I will discuss applications of single and multi-component quantum fluids and classical two-fluid systems in strong gradient magnetic fields.

• Thematic program: Models in Plasmas, Earth and Space Science (2019/2020)
• Event: "Experimental and Numerical Quantum Cosmology". (2020)

Atomic physics experiments, based on hot vapors or laser-cooled atomic samples, may be useful to simulate some astrophysical problems, where radiation pressure, radiative transport or light amplification are involved. I will present some ongoing experimental efforts in Nice and discuss spontaneous self-organisation with light-induced long-range forces.

• Thematic program: Models in Plasmas, Earth and Space Science (2019/2020)
• Event: "Experimental and Numerical Quantum Cosmology". (2020)

• Thematic program: Models in Plasmas, Earth and Space Science (2019/2020)
• Event: "Experimental and Numerical Quantum Cosmology". (2020)

We develop semantically-oriented calculi for the cube of non-normal modal logics and some deontic extensions. The calculi manipulate hypersequents and have a simple semantic interpretation. Their main feature is that they allow for direct countermodel extraction. Moreover they provide an optimal decision procedure for the respective logics. They also enjoy standard proof-theoretical properties, such as a syntactical proof of cut-admissibility.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

In this talk I will present a nested sequent system for a combination of (non-normal) monotone modal logic M and normal modal logic K. The system is fully internal, can be used for proof search, and is suitable for countermodel construction. I will also consider some deontic extensions and present a prototype implementation.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

Defeasible Logic is a simple practical computationally oriented (sceptical) non-monotonic formalism that proved (i) to be flexible to capture different facets of non-monotonic reasoning and (ii) to be extensible. The logic is based on a constructive proof theory. We are going to show to use the proof theory to extend the logic with modalities, and to capture some aspects of substructural logic. We also show how to use some of these features to address some paradoxes of deontic logic.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

In this talk we look at how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. As a consequence of the extraction process, each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

As follows from their name, tree-hypersequents (also known as nested sequents) were created to represent the tree structure of underlying Kripke models. While this approach works well on modal and intermediate logics complete w.r.t. many types of tree-like frames, it is not directly suited to encode quantitative restrictions on these frames, e.g., bounded depth and/or bounded number of children per node. In order to capture these restrictions, we add the injectivity condition to nested sequents requiring different sequent nodes to correspond to distinct worlds in the underlying Kripke model. The downside is the loss of the formula interpretation. On the plus side, we show how the injective nested sequents can be used to constructively prove the Craig interpolation property for all interpolable intermediate logics strictly between the intuitionistic and classical propositional logics that are complete with respect to tree-like models, i.e., Smetanich logic (also known as the logic of here and there), the greatest semiconstructive logic, logic BD_2 of bounded depth 2, and Gödel logic. For the last one, we obtain a stronger form of interpolation called Lyndon interpolation.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

I shall expose recent advances on exploring the equaltional theories of the residuated lattices Q(C) made of join-continuous endofunctions of a complete chain C. On one side, when investating congruences, we observed that the number of idempotents in the residuated lattice Q({0,1,...,n}) is the 2n+1-th Fibonacci number. Our proof yields a combinatorial interpretation of results due to Howie and Laradji-Umar. If C is a finite chain or the interval [0,1] of the reals, Q(C) is an involutive residuated lattice. Generalizing this fact, we shall present the following result : for a complete lattice L, the residuated lattice Q(L) of join-continuous endofunctions of L is involutive if and only of L is a completely distributive lattice. Thus, the step from ILL to MALL requires, for those residuated lattices, also a classical structure on the additives. It also holds that Q(L) is an involutive mix residuated lattice if and only if L is a complete chain.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

The logic of STIT (`seeing to it that') is an agency logic for reasoning about agents that make choices at certain moments in time. This class of modal logics has received considerable attention in the past decades with formal application in epistemic-, legal-, and deontic reasoning. Furthermore, in relation to the increasing development of autonomous systems assisting and interacting with humans, the need for automated normative reasoning with STIT logics has been stressed in the literature. Our present research addresses this issue. In this talk we will set out the concrete aims of our STIT project and discuss some of the results obtained so far. We will first provide an introduction to the logic of STIT and discuss our recently proposed Temporal Deontic extension STIT. Second, we provide labelled sequent calculi for the class of multi-agent STIT logic with limited choice axioms and show how these calculi can be refined with the use of propagation rules, enabling us to reduce the structure of sequents and to make the proofs more compact. For the class of refined calculi we obtain automated proof-search and counter-model extraction. We will conclude by discussing some open problems.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

TBA

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

Many substructural, intermediate and modal logics have found cut-free presentations in the hypersequent calculus. We demonstrate that for many such logics, this cut-freeness at the level of hypersequents also implies completeness with respect to a sequent system where only cuts of a certain shape are allowed. The restriction on the cuts thus obtained is often strong enough to allow for proofs of metalogical properties such as decidability, or embeddability into a weaker base logic. Our method also allows for a new proof of the fact that the modal logic S5 has a sequent calculus in which only analytic cuts are needed.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

In this talk we discuss proof translations between labelled and label-free calculi for the logic of Bunched Implications (BI). We first consider the bunched sequent calculus LBI and define a labelled sequent calculus, called GBI, in which labels and constraints reflect the properties of a specifically tailored Kripke resource semantics of BI with two total resource composition operators and explicit internalization of inconsistency. After showing the soundness of GBI wrt our specific Kripke frames, we show how to translate any LBI-proof into a GBI-proof. Building on the properties of that translation we devise a tree property that every LBI-translated GBI-proof enjoys. We finally show that any GBI-proof enjoying this tree property (and not only LBI-translated ones) can systematically be translated to an LBI-proof.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

Over the course of more than two millennia, the philosophical school of Mimamsa has thoroughly discussed and analyzed the prescriptive portion of the Vedas, the sacred texts of Hinduism, in order to make sense of it as a consistent corpus of rules. We present a formalization of the deontic system applied by Mimamsa authors for resolving conflicts between normative statements by giving preference to the more specific ones. Finally, we show how to use the resulting system to provide a better understanding of these philosophical texts.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem - in our case by a Minsky machine - is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

This paper focuses on a globally sound but possibly locally unsound analytic sequent calculus for the quantifier macro Q. It is demonstrated that no locally sound analytic representation exists.

• Thematic program: Informatics and Logic (2019/2020)
• Event: Workshop on "TICAMORE - Translating and Discovering Calculi for Modal and Related Logics" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

• Thematic program: Mathematics, Materials and Electro-Magnetism (2019/2020)
• Event: Working Group on "Micromagnetics of Permanent Magnets" (2019)

I will present a new normal form transformation decomposing the dynamics of some nonlinear Hamiltonian systems into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolves according to a time dependent linear Hamiltonian system. We then control the global dynamics by using poly- nomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. We will see how this procedure allows us to prove that, for almost any mass, small and smooth solutions of the nonlinear wave equation on Td of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. This is a joint work with Erwan Faou and Benoit Grebert.

This lecture is devoted to the study of the Hele-Shaw equation, based on a joint work with Nicolas Meunier and Didier Smets. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancel- lations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equa- tions in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapounov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The Euler-Korteweg system is a dispersive perturbation of the usual compress- ible Euler equations that includes the effect of capillary forces. For small ir- rotational initial data, global well-posedness is known to hold in dimension at least three. In this talk we discuss the case of small initial data with non zero vorticity, where the dispersive system becomes a coupled dispersive-transport system. The main result is that the time of existence only depends on the size of the initial vorticity.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In this presentation, we consider the nonlinear Schr¨odinger equation with loga- rithmic nonlinearity (logNLS in short). We mostly focus on the focusing case which presents a very special Gaussian stationary solution, called Gausson, which is orbitally stable. In fact, more generally, it has been shown that every Gaussian data remains Gaussian through the flow of logNLS, and this feature gives rise to (almost) periodic solutions in the focusing case, called breathers. The main result of this talk addresses the existence of multi-solitons, i.e. solu- tions to logNLS which behaves like the sum of several solitons (i.e. Gaussons here) for large times, in dimension 1. This kind of result is rather usual for dispersive equations with polynomial-like nonlinearity, and our proof is directly inspired from the usual proof with energy techniques. The main difficulty is the fact that the energy cannot be linearized as one would want, at least not ev- erywhere. Furthermore, some new and surprising features appear in this result: the convergence is in H1 and (H1) with a rate faster than exponential, and there is no need for a large enough relative speed (non-zero is sufficient).

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In a series of papers with Sylvie Benzoni-Gavage (and, depending on papers, Pascal Noble or Colin Mietka), we have studied both co-periodic stability and modulation systems for periodic traveling waves of a rather large class of Hamil- tonian partial differential equations that includes quasilinear generalizations of the Korteweg–de Vries equation and dispersive perturbations of the Euler equa- tions for compressible fluids, either in Lagrangian or in Eulerian coordinates. All characterizations are derived in terms of the Hessian matrix of the action integral of profile equations, a finite-dimensional object. In the present talk, with this in mind, we shall discuss the consequences of the recently obtained expansions of this matrix in two asymptotic regimes, namely the zero-amplitude and the zero-wavelength limits.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The binormal flow is a model for the dynamics of a vortex filament in a 3-D in- viscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schr¨odinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results presented in this talk is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity. This is a joint work with Luis Vega.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The linear stability of homogenous shear flows between two rigid walls is a clas- sical problem that goes back to Rayleigh (1880). Among other things, Rayleigh was able to show that a shear flow with no inflection points is linearly stable. The generalisation of this stability criterion to the free-surface setting is not straightforward and was established much later by Yih (1971) (under certain restrictions) and, more recently, Hur & Lin (2008). In the case when a shear flow with a free surface is modelled by constant vorticity layers, no stability criterion is known. As a first step in this direction we consider the stability analysis of a bilinear shear current and establish a criterion for the stability of the flow. The effect of density stratification on the stability of the flow will also be investigated.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The Serre-Green-Naghdi system is a fully nonlinear and weakly dispersive model for the propagation of surface gravity waves. It enjoys many good theoretical properties, including a robust well-posedness theory for the initial-value prob- lem, and a Hamiltonian structure. It is however not so suitable for practical use, as standard numerical strategies involve the costly inversion of an elliptic operator at each time step. N. Favrie and S. Gavrilyuk proposed a novel strat- egy for efficiently producing approximate solutions, by introducing a “relaxed” first-order quasilinear system of balance laws, depending on additional unknows and a free parameter. The claim is that in the singular limit when the param- eter goes to infinity, solutions of the relaxed system approach solutions of the Serre-Green-Naghdi system. We will discuss a rigorous analysis. It differs from standard results due to the presence of an additional parameter (describing the shallowness of the flow) and order-zero source terms which become dominant when the shallowness parameter goes to zero.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

This is a jointwork with Thomas Kappeler. Using the Lax pair structure for the Benjamin-Ono equation with periodic boundary conditions, we construct a global system of Birkhoff coordinates on the phase space of real valued square integrable functions with average 0 on the torus, including a characterisation of finite gap potentials. Among consequences, we infer almost periodicity of all trajectories, identification of traveling waves and construction of periodic in time solutions with low regularity.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

We discuss numerical methods to construct solutions to nonlinear dispersive PDEs on the whole real line, and this for initial data which are slowly decreasing towards infinity or just bounded there. As an example we discuss the transverse stability of the Peregrine solution in the 2d nonlinear Schrodinger equation.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In this work we will look at the focusing Davey-Stewartson equation from two different angles, using advanced numerical tools. As a nonlinear dispersive PDE and a generalisation of the non-linear Schr¨odinger equation, DS possesses solutions that develop a singularity in finite time. We numerically study the long time behaviour and potential blow-up of solutions to the focusing Davey-Stewartson II equation for various initial data and propose a conjecture describing the blow up rate and solution profiles near the singularity. Secondly, DS is an integrable system and can be studied as an inverse scat- tering problem. Both the forward and inverse scattering transformation in this case are reduced to a d-bar system which plays the role that Riemann-Hilbert problems play in one dimensional problems. We will present numerical solutions for Schwartzian and compactly supported potentials. Further, to complement numerics, we will discuss analytical considerations to handle asymptotic be- haviour. In all studied cases we use spectral methods and achieve machine pre- cision. Based on joint works with Christian Klein and Ken McLaughlin

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

This is a joint work with Christophe Cheverry. We study high frequency so- lutions of nonlinear hyperbolic equations for time scales at which dispersive and nonlinear effects can be present in the leading term of the solution, on a model stemming from strongly magnetized plasmas or nuclear magnetic reso- nance experiments. We show how the produced waves can accumulate during long times to produce constructive and destructive interferences which, in the above contexts, are part of turbulent effects.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

TBA

• Thematic program: Stochastics and Finance (2019/2020)
• Event: Working Group on "Infinite Dimensional Polynomial Processes" (2019)

TBA

• Thematic program: Stochastics and Finance (2019/2020)
• Event: Working Group on "Infinite Dimensional Polynomial Processes" (2019)

TBA

• Thematic program: Stochastics and Finance (2019/2020)
• Event: Working Group on "Infinite Dimensional Polynomial Processes" (2019)

TBA

• Thematic program: Stochastics and Finance (2019/2020)
• Event: Working Group on "Infinite Dimensional Polynomial Processes" (2019)

TBA

• Thematic program: Stochastics and Finance (2019/2020)
• Event: Working Group on "Infinite Dimensional Polynomial Processes" (2019)

Advances in tumor immunology now calls for a novel understanding of the immunological consequence of standard cancer therapy. At the same time the expression of proteins mediating immunogenic cell death should have a positive predictive and prognostic impact. This molecular understanding of the disease will allow a more rational design of immunomodulating drugs and standard therapy. Murine models clearly indicate that irradiation induced DNA damage can stimulates the innate and adaptive immune system. However, there is little evidence that irradiation leads to apiscopal effects in the clinic. We here show that neoadjuvant irradiation applied in rectal cancer patients induces the polarization of tumor associated M2-like macrophages to an M1-like phenotype in surgical resection specimen. Ex vivo primary cultures and organotypic assays were used to better dissect this re-polarization. Using exvivo cultures we further show that the shift of irradiation-induced macrophage polarization could be mediated by exosomes. Those data clearly indicate that radiotherapy induced DNA damage using 25 Gy actively stimulates the innate immune system. This pro-inflammatory effect of radiotherapy might now be complemented by immunomodulating drugs modulating the adaptive part of the immune system. In contrast, when analyzing the prognostic and predictive impact of spontaneous DNA damage and associated pathways in colorectal liver metastases we demonstrate that DNA damage had a strong negative impact on response to neoadjuvant applied chemotherapy but also on disease free and overall survival. Spontaneous DNA damage was not associated with an induction of the innate immune response in this setting and inversely correlated with infiltrates of CD8+ or CD45RO+ cells. This calls for a more detailed understanding of spontaneous DNA damage induced pathways in colorectal liver metastases as their blockade might enhance prognoses.

From protein interactions to signal transduction, from metabolism to the nervous system: Virtually all processes in health and disease rely on the careful orchestration of a large number of diverse individual components ranging from molecules to cells and entire organs. Networks provide a powerful framework for describing and understanding these complex systems in a wholistic fashion. They offer a unique combination of a highly intuitive, qualitative description, and a plethora of analytical, quantitative tools. In my presentation, I will introduce three ongoing projects of my group, each highlighting a different aspect of how network science can help us understand the pathobiological processes of human disease: First, I will sketch out how protein-protein interaction networks can be understood as maps to investigate relationships between diseases. Second, I will discuss how drug-drug interaction networks can be used to identify basic principles of the cellular response to multiple perturbations. Lastly, I will present our vision of a virtual reality platform for the next generation of network-based data integration and exploration.

One of the most important cellular behaviors is cell crawling migration. It is observed in many cellular systems both in culture and in vivo, and involved in many essential physiological or pathological processes (wound healing, embryonic development, cancer metastasis etc). As in the last decade adhesion-independent migration has been observed in confining environement and has emerged as a possibly common migration mode, we propose a simplified 2D model for focal adhesion-free cell migration: A cell is modeled through its membrane represented as a set of connected springs which undergo internal pressure forces. The renewal of the actin network is modelled by creation/suppression of springs in the membrane, and we suppose that a cell generates internal counter-forces compensating mass displacement due to membrane renewal. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for cell motility in confined environment.

An engineer designing a communication system would use few distinct signaling components while ensuring that the output of each component is highly accurate. However, natural evolution came up with a different solution: cells have many interconnected, cross- reactive components that individually produce noisy signals. Why? In the talk, I will present the perspective of mathematical information-theory at the two intriguing properties of cellular signaling pathways: noisiness and cross-talk. Specifically, I will discuss their (i) evolutionary origins; (ii) implications for interpretation of single cell data; and (iii) consequences for the design of therapeutic interventions in signaling.

Understanding how tissues develop and regulate their growth is crucial in biology. Both proliferation and regulation of cells growth are fundamental for the development of healthy tissue in animals and plants, as well as for the progression of tumours. In pseudostratified epithelia, the organisation of the nuclei and their movement inside the tissue influence the final architecture of the tissue and impact growth. In particular, nuclei move along the apical/basal axis during the inter-kinetic phases of the cell cycle. This movement is called the inter-kinetic nuclear movement. Because pseudostratified epithelia have a high density of nuclei, their movement is likely to be influenced by the crowing inside the tissue. We developed an Individual-based model for the interkinetic nuclear movement in pseudostratified epithelia based in a minimisation framework. The model focuses is placed on the nuclei and their deformation. We study the influence of crowding the specific case of the Imaginal Disc of Drosophila and tuned the model with biological data. We then show that the crowding increases the cell cycle duration, resulting in the slow down of growth.

In cancer research most efforts are devoted on the decipher of the IntraTumoral Heterogeneity (ITH). In ITH the action of the evolutionary forces of mutation and selection are essential to determinant the tumor progression, diagnosis and treatment. ITH gives rise to cancer cell populations with distinct genotypic and metabolic characteristics contributing to the failure of cure, by initiating phenotypic diversity and enabling more aggressive and drug resistant clones. I will present multi-scale models of cancer linking the tumor growth to the intracellullar signalling and metabolic events to genomic profiles. The models consider several heterogenous omics data (metabolomics, proteomics, transcriptomics, genomics) to investigate the ITH associated with different genomic and metabolic traits.

Currently, immunotherapy with checkpoint inhibitor antibodies is revolutionizing clinical oncology even allowing cure of highly aggressive cancer types like melanoma and lung cancer. However, response to these immunotherapies is restricted to patient subgroups and currently conclusive predictive biomarkers are not available. Classically, anticancer metal drugs are considered to target predominantly nucleic acids, hence killing cancer cells by inducing genomic damage and apoptotic cell death. However, during the last years it became clear that metal drugs are not pure cytotoxic agents, but might also strongly interact with the fidelity of anticancer immune responses. Central underlying mechanisms include upregulation of cancer cell immunogenicity or depletion of regulatory immune cell compartments1. As one example, we have found that an intraperitoneal colon cancer model can be cured when combining oxaliplatin with bacterial ghosts as adjuvants2. Bacterial ghosts are empty envelopes of gram-negative bacteria with a distinct immune-stimulatory potential. In contrast, oxaliplatin alone only retarded tumor growth. Interestingly, animals cured by this immunochemotherapy approach were vaccinated against the original cancer cells making regrowth of the tumor graft impossible. As this vaccination effect was entirely depending on the presence of activated T cells, induction of an immunogenic cell death by oxaliplatin supported by innate immune activation via the adjuvant can be anticipated. This hypothesis was proven be induction of endoplasmic reticulum (ER) stress, calreticulin cell surface exposure, as well as HMGB1 and ATP release be the combination-treated cancer cells. A platinum(IV) prodrug of oxaliplatin targeted for tumor-specific activation based on albumin binding was able to cure CT26 murine colon cancer even without additional adjuvant in immunocompetent but not severe combined immunodeficient (SCID) mice3. The question arises whether mathematical modelling of at least parts of the complex interplay between DNA damage and immune activation by anticancer platinum drugs would be conceivable.

A number of studies have demonstrated that the disordered process of angiogenesis occurring in malignant tumours produces stochastic variations in blood flow leading to cycles of perfusion, cessation of flow, and then re-perfusion. This produces corresponding fluctuations in environmental conditions that include the concentration of nutrients, such as oxygen and glucose. In order to support a deeper understanding of the adaptive role of spontaneous phenotypic variations in cancer cell populations exposed to fluctuating environments, we consider a system of non-local partial differential equations modelling the evolutionary dynamics of two competing populations in the presence of periodically oscillating nutrient levels. Exploiting the analytical tractability of our model, we study the long-time behaviour of the solutions to obtain a detailed mathematical depiction of evolutionary dynamics. Our analytical results formalise the idea that when nutrient levels experience small and slow periodic oscillations, and thus environmental conditions are relatively stable, it is evolutionarily more efficient to rarely undergo spontaneous phenotypic variations. Conversely, under relatively large and fast periodic oscillations in the nutrient levels, which lead to alternating cycles of starvation and nutrient abundance, higher rates of spontaneous phenotypic variations can confer a competitive advantage, as they may allow for a quicker adaptation to changeable environmental conditions. In the latter case, our results indicate that higher levels of phenotypic heterogeneity are to be expected compared to those observed in slowly fluctuating environments. Finally, our results suggest that bet-hedging evolutionary strategies, whereby cancer cells switch between antithetical phenotypic states, can naturally emerge in the presence of relatively large and fast nutrient fluctuations leading to drastic environmental changes.

Chromatin immunoprecipitation followed by sequencing (ChIP-seq) is widely used in the global investigation of protein-DNA interactions. One of its main applications is the analysis of differential chromatin binding patterns of the proteins of interest in varying biological states. While various algorithms can be used to quantitatively compare ChIP-seq datasets, different computational tools apply different normalization strategies, which can strongly influence the results of the analyses. Applying inappropriate normalization can lead to erroneous outcomes, and the performance of different tools can strongly depend on the nature of the investigated dataset. Therefore it is hard to choose the most appropriate differential ChIP-seq tool. To overcome this limitation, we systematically assessed available tools for differential ChIP-seq analysis to provide recommendations which tools to use for different biological scenarios and data types. We created standardized reference datasets by in-silico simulation of ChIP-seq data to represent different biological scenarios, including global reduction of genomic regions in one sample versus the other, but also up- and down-regulation of equal proportions of genomic regions in both samples. We used these scenarios to evaluate the performance of 24 computational tools for differential ChIP-seq analysis. We found enormous differences in precision and recall across differential ChIP-seq analysis tools. The performance was strongly dependent on the sizes and shapes of simulated peaks as well as on the regulation scenario. We are currently extending these findings to publicly available and unpublished experimental ChIP-seq datasets. Our analysis provides unbiased recommendations which tools to use for particular biological scenarios. The application of appropriate analysis tools will greatly improve the outcomes of ChIP-seq studies, and will thus contribute to improved identification of molecular mechanisms.

Molecular descriptor (2D) and three dimensional (3D) shape based similarity methods are widely used in ligand based virtual drug design. In the present study pairwise structure comparisons among a set of 4858 DTP compounds tested in the NCI60 tumor cell line anticancer drug screen were computed using chemical hashed fingerprints and 3D molecule shapes to calculate 2D and 3D similarities, respectively. Additionally, pairwise biological activity similarities were calculated by correlating the 60 element vectors of pGI50 values corresponding to the cytotoxicity of the compounds across the NCI60 panel. Subsequently, we compared the power of 2D and 3D structural similarity metrics to predict the toxicity pattern of compounds. We found that while the positive predictive value and sensitivity of 3D and molecular descriptor based approaches to predict biological activity are similar, a subset of molecule pairs yielded contradictory results. By simultaneously requiring similarity of biological activities and 3D shapes, and dissimilarity of molecular descriptor based comparisons, we identify pairs of scaffold hopping candidates displaying characteristic core structural changes such as heteroatom/heterocycle change and ring closure. Attempts to discover scaffold hopping candidates of mitoxantrone recovered known Topoisomerase II (Top2) inhibitors, and also predicted new, previously unknown chemotypes possessing in vitro Top2 inhibitory activity.

The main goal of this talk is to present examples of how mathematical modeling and AI may help clinicians following the evolution of cancer. The first example uses machine learning to evaluate the efficacy of neoadjuvant chemotherapy of softtissue sarcoma. Standard of care for advanced stages (grade 3) is the following: neoadjuvant chemotherapy (6 cycles), curative surgery and then adjuvant radiotherapy. Unfortunately, for some patients, chemotherapy does not improve the situation. In clinical routine, two MR exams are performed on patients: one before the chemotherapy and one after two cycles. Using a retrospective study of more than 60 patients from Institut Bergonié, we investigate whether the differences between these two exams may be correlated with response to chemotherapy. For this matter a radiomics approach is used with novel handcrafted features specific to the disease. On the cohort, the results we obtain are better than state of the art. In the second example, we try to evaluate the efficacy of tyrosine kinase inhibitors (TKI) for patients with EGFR mutated Non-Small Cell Lung Carcinoma. Patients almost always end up relapsing. Our goal is to analyze if an insight on this relapse may be obtained from the early response to treatment. We built a mathematical model — based on a set of PDE - of the response to TKI. This model is personalized for each patient of a retrospective cohort from Institut Bergonié. For the patient-specific model, we compute a novel marker that we show to be correlated with risk of relapse. Finally, a new data assimilation technique will be presented that is able to recover patient-specific parameters of a PDE model of growth of brain metastases. It may be used to predict the evolution of these metastases.

In high grade serous ovarian cancer patients with peritoneal involvement have an unfavorable outcome and would benefit from targeted therapies. In the last years we comprehensively described two types of peritoneal tumor spreading, miliary, with many millet sized tumor nodules in the peritoneal cavity, and non-miliary, with few larger and exophytically growing tumors. The former showed significant shorter survival, therefore we aimed to find a druggable target against miliary peritoneal metastasizing. We constructed a planar – scale free and small world – co-association gene expression network from RNAsequencing data using mutual information as the association measure, defined sub-clusters with multiscale clustering, and searched for sub-clusters with hub genes up-regulated in miliary tumors. A subcluster of 38 genes and Nectin 4 as hub-gene was among the highest significant up-regulated sub-clusters. Using the genes of this sub-cluster for a gene signature we validated the impact on survival with six publicly available expression datasets. Protein expression and impact on survival of Nectin 4 was validated via immunohistochemistry and correlated to other omics and medium-dimensional data. Results were condensed to a network and used for biological interpretation of the impact of Nectin 4 on peritoneal ovarian cancer metastasizing. An anti-Nectin 4 antibody with a linked antineoplastic drug – already used in clinical trials for cancer treatment – could be a promising candidate for a targeted therapy in patients with miliary peritoneal involvement.

Colorectal cancer (CRC) is one of the most common causes of cancer-related mortality worldwide. Most cases of deaths result from metastases, assumed to be shed, in many cases, before disease detection. Providing reliable predictions of the metastases' growth pattern may help planning treatment. Available mathematical tumor growth models rely mainly on primary tumor data, and rarely relate to metastases growth. The aim of this talk was to explore CRC lung metastases growth patterns. We used data of a metastatic CRC patient, for whom ten lung metastases were measured while untreated by seven serial computed tomography (CT) scans, during almost three years. Three mathematical growth models – Exponential, logistic and Gompertzian – were fitted to the actual measurements. Goodness of fit of each of the models to actual growth was estimated using different scores. Factors affecting growth pattern were explored: size, location and primary tumor resection. Exponential growth model demonstrated good fit to data of all metastases. Logistic and Gompertzian growth models, in most cases, were overfitted and hence unreliable. Metastases inception time, calculated by backwards extrapolation of the fitted growth models, was 8-19 years before primary tumor diagnosis date. Three out of ten metastases demonstrated enhanced growth rate shortly after primary tumor resection. Our unique data provide evidence that exponential growth of CRC lung metastases is a legitimate approximation, and encourage focusing research on short-term effects of surgery on metastases growth rate.

DNA methylation is known to have a major impact on the protein biosynthesis of tissues. Those epigenetic modifications could theoretically also be used for tissue classification. In this study we hypothesized that machine learning algorithms could be applied to distinguish between different tissue samples.

How combination therapies can reduce the emergence of cancer resistance? Can we exploit intratumoral competition to modify the effectiveness of anti-cancer treatments? Bearing these questions in mind, we present a mathematical model of cancer-immune competition under therapies. The model consists of a system of differential equations for the dynamics of two cancer clones and T-cells. Comparisons with experimental data and clinical protocols have been performed. In silico experiments confirm that the selection of proper infusion schedules plays a key role in the success of anti-cancer therapies. The outcomes of protocols of chemotherapy and immunotherapy (separately and in combination) differing in doses and timing of the treatments are analyzed. In particular, we highlight how exploiting the competition between cancer populations seems to be an effective recipe to limit the insurgence of resistant populations. In some cases, combination of low doses therapies could yield a substantial control of the total tumor population without imposing a massive selective pressure that would suppress the sensitive clones leaving unchecked the clonal types resistant to therapies.

In this study, we propose a novel direct numerical method with computational cost to simulate the wave equation in the seismic inverse problem. It based on the fact that the computation of the entire wave filed may not be necessary here. Thus, we only need to evaluate the wave equation around the waveform of interest and the computational cost is significantly saved here.

The dynamics of the early universe and black holes are deeply linked to the interplay between general relativity and quantum fields. The essential physical processes occur in situations that are hard to observe and impossible to experiment with: when gravitational interactions are strong and/or when quantum effects are important. Analog (quantum) simulators, utilising the analogy between the dynamics of perturbations in classical or quantum fluids and relativistic fields in a curved space-time metric, enable us to study these processes in controlled laboratory setups. I will give an introduction to the current questions, opportunities, and challenges concerning analog simulations in the context of early universe cosmology, focusing, in particular, on cosmic inflation. It is our current understanding that, during inflation, quantum fluctuations are stretched to cosmic scales during the rapid expansion of space-time, yielding the seed for the large scale structure formation in our universe. Analog simulations give direct experimental access to the field dynamics in these extreme conditions, enabling us to study the underlying processes, like mode-freezing, particle creation, quantum-classical transition, and signature changes in the space-time metric, in detail.

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• Thematic program: Mathematics for Materials and Micromagnetism (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

The aim of this talk is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the "universality" of the Onsager exponent, larger than 1/3, concerning the regularity of the solutions - space of Hölder continuous functions with the above exponent, that guarantees the conservation of the generalized entropy; regardless of the structure of the genuine nonlinearity in the underlying system.

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued - strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. Following result of Yann Brenier, Camillo De Lellis and Laszlo Szekelyhidi Jr. for incompresible Euler equation, this property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system, polyconvex elastodynamics et al. In my talk I will concentrate on results concerning general conservation laws. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems. Following earlier common result with Eduard Feireisl, Piotr Gwiazda and Emil Wiedemann for compresible Navier-Stokes system, we develop the concept of dissipative measure-valued solution to general hyperbolic systems. The talk is based on joint results with Piotr Gwiazda and Ondrej Kreml.

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

This talk proposes a derivation of the Vlasov-Navier-Stokes system used in the modeling of "thin" aerosol flows from a system of Boltzmann equations for a binary gas mixture involving the propellant gas and the dispersed phase in the aerosol. This derivation is formal, in the sense of the program for deriving fluid dynamic limits of the Boltzmann equation laid out in [C. Bardos - F. Golse - C.D. Levermore: J. Stat. Phys. 63 (1991), 323-344].

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

I will discuss the construction of wild weak solutions to the Navier-Stokes equation which are smooth on the complement of a thin set of times (with Haursdorff dimension strictly less than 1). This is based on joint work with T. Buckmaster and M. Colombo.

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

It is a notorious and classical problem whether Leray solutions of the Navier-Stokes equations converge to a solution of the Euler equations, as viscosity tends to zero. The problem is only well-understood in the case that the Euler solution is smooth and there are no physical boundaries. If one (or both) of these requirements are violated, the problem is still largely open. We discuss two specific situations: First, we prove a version of Onsager's conjecture in bounded domains that gives rise to a statement on the viscosity limit and the absence of anomalous dissipation (joint work with C. Bardos and E. S. Titi). Secondly, we discuss the viscosity limit problem for the (non-smooth) shear flow, also departing from work with Bardos and Titi; we investigate in particular the question what happens when the initial data is not exactly fixed along the viscosity sequence (in progress).

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: Workshop on "Mathematical Amelioration in Fluid Dynamics" (2018)

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Working Group on "Dipolar Bose Einstein Condensates: Experiments and numerical simulations" (2018)

Over the last years, finite element methods based on operator-adapted approximating spaces have been developed in order to better reproduce physical properties of the analytical solutions, and to enhance stability and approximation properties. They are based on incorporating a priori knowledge about the problem into the local approximating spaces, by using trial and/or test spaces locally spanned by functions belonging to the kernel of the differential operator (Trefftz spaces). These methods are particularly popular for wave problems in frequency domain. Here, the use of oscillating basis functions allows to improve the accuracy vs. computational cost, with respect to standard polynomial finite element methods, and breaks the strong requirements on number of degrees of freedom per wavelength to ensure stability. In this talk, the basic principles of Trefftz finite element methods for time-harmonic wave problems will be presented. Trefftz methods differ from each other by the way interelement continuity conditions are imposed. We will focus on discontinuous Galerkin approaches, where the approximating spaces are made of completely discontinuous Trefftz spaces, and on the recent virtual element framework.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

Electric Impedance Tomography (EIT) is a medical imaging technique that uses the response to voltage difference applied outside the body to reconstruct tissue conductivity. As different organs have different impedance, this technique makes it possible to produce images of the body without exposing the patient to potentially harmful radiation. In mathematical terms, EIT is what is a nonlinear inverse problem, whereby data inside a given domain is recovered from data on its boundary. Such problems also belong to the area of Integrable Systems, which deals with nonlinear problems for which analytic solutions can be found, thus providing us with a mathematical framework for reconstructing images from the electrical information created by EIT. I will discuss the design of numerical algorithms based on spectral collocation methods that address D-bar problems found in both integrable systems and medical imaging. Successfully implementing these methods in EIT on modern computing architectures should allow us to achieve images with much higher resolutions at reduced processing times.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

We show that the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $$ \partial_tu-D_x^{\alpha}\partial_xu+u\partial_xu=0 \, ,$$ with $0<\alpha \le 1$, is locally well-posed in $H^s(\mathbb R)$ for $s>s_\alpha: = \frac 32-\frac {5\alpha} 4$. As a consequence, we obtain global well-posedness in the energy space $H^{\frac{\alpha}2}(\mathbb R)$ as soon as $\frac\alpha 2> s_\alpha$, i.e. $\alpha>\frac67$.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

Complex normal coordinates for integrable PDEs on the torus can be viewed as 'nonlinear Fourier coefficients'. Based on previous work we construct near an arbitrary finite gap potential a real analytic, 'nonlinear Fourier transform' for the KdV equation having the following two main properties: (1) Up to a remainder term, which is smoothing to any given order, it is a pseudodifferential operator of order 0 with principal part given by the Fourier transform. (2) It is canonical and the pullback of the KdV Hamiltonian is in normal form up to order three. Furthermore, the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions, i.e., periodic multisolitons, of the KdV equation under small, quasi-linear perturbations. This is joint work with Riccardo Montalto.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

The Szegö equation is an integrable model for lack of dispersion on the circle. An important feature of this model is the existence of a residual set --- in the Baire sense--- of initial data leading to unbounded trajectories in high Sobolev norms. It is therefore natural to study the effect of a weak damping on such a system. In this talk I will discuss the damping of the lowest Fourier mode, which has the specificity of saving part of the integrable structure. Somewhat surprinsingly, we shall show that such a weak damping leads to a wider set of unbounded trajectories in high Sobolev norms. This is a jointwork in collaboration with Sandrine Grellier.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

This talk reports on joint work with Robert Jenkins, Jiaqi Liu, and Catherine Sulem. The derivative nonlinear Schr\"{o}dinger equation (DNLS) is a completely integrable, dispersive nonlinear equation in one space dimension that arises in the study of circularly polarized Alfv\'{e}n waves in plasmas, and admits soliton solutions. In 1978, Kaup and Newell showed that the DNLS is completely integrable, and in the 1980's, J.-H. Lee used the Beals-Coifman approach to inverse scattering to solve the DNLS. In the work to be described, drawing on recent advances in the Riemann-Hilbert formulation of inverse scattering due to Dieng-McLaughlin (2008) and Borghese-Jenkins-McLaughlin (2017), we use the inverse scattering formalism to show that, for a spectrally determined generic set of initial data, the solution decomposes into the sum of 1-soliton solutions with calculable phase shifts plus radiation.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

Boiti-Pempinelli-Pogrebkov's inverse scattering theories on the KPII equation provide an integrable approach to solve the Cauchy Problem and the stability problem of the KPII equation for perturbed multisoliton solutions. In this talk, we will present rigorous analysis for the direct scattering theory of perturbed KPII one line solitons, the simplest case in Boiti-Pempinelli-Pogrebkov's theories. Namely, for generic small perturbation of the one line soliton, the existence of the eigenfunction is proved by establishing uniform estimates of the Green function and the Cauchy integral equation for the eigenfunction is justified by nonuniform estimates of the spectral transform. Difficulties and outlooks for the inverse problem will be discussed as well.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

We are concerned with 1D scattering problems related to quantum transport in (tunneling) diodes. The problem includes both oscillatory and evanescent regimes, partly including turning points. We shall discuss the efficient numerical integration of ODEs of the form epsilon^2 u" + a(x) u = 0 for 0 < epsilon << 1 on coarse grids, but still yielding accurate solutions. In particular we study the numerical coupling of the highly oscillatory regime (i.e. for given a(x) > 0 ) with evanescent regions (i.e. for a(x) < 0 ). In the oscillatory case we use a marching method that is based on an analytic WKB-preprocessing of the equation. And in the evanescent case we use a FEM with WKB-ansatz functions. We present a full convergence analysis of the coupled method, showing that the error is uniform in epsilon and second order w.r.t. h, when h = O(epsilon^1/2). We illustrate the results with numerical examples for scattering problems for a quantum-tunnelling structure. The main challenge when including a turning point is that the solution gets unbounded there as epsilon -> 0. Still one can obtain epsilon-uniform convergence, when h = O(epsilon^7/12).

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

We study numerically the stability of solitons and a possible blow-up of solutions in dispersive PDEs of the family of Kortweg-de Vries and nonlinear Schr\"odinger equations. The biow-up mechanism in the $L^2$ critical and supercritical case is studied.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

After recalling the known results on the KP I and KP II equations, we survey some open problems on the KP equations, both from the PDE and IST aspects, and also on some relevant KP type equations.

• Thematic program: Mathematical methods for many body physics (2018/2019)
• Event: Workshop on "Dispersion and Integrability" (2018)

TBA

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

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• Thematic program: Models in Plasmas, Earth and Space Science (2018/2019)
• Event: 11th Plasma Kinetics Working Group Meeting (2018)

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

In this talk, I focus on current biological problems and on how to use mathematical modeling to analyze a variety of pressing questions arising from oncology, developmental pattern formation and population ecology. I first discuss novel mathematical models for cancer growth dynamics and heterogeneity. These studies rely on evolutionary principles and shed light on 3D hepatic tumor dynamics, spatial heterogeneity and tumor invasion, and single cancer cell responses to antimitotic therapies. We also develop mathematical models that quantitatively demonstrate how the interplay between non-genetic instability, stress-induced adaptation, and selection leads to the transient and reversible phenotypic evolution of cancer cell populations exposed to therapy. Finally, we study control techniques for optimal therapeutic administration.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

TBA

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

As the spinal cord grows during embryonic development, an elaborate pattern of molecularly distinct neuronal precursor cells forms along the DV axis. This pattern depends both on the dynamics of a morphogen-regulated gene regulatory network, and on tissue growth. We study how these processes are coordinated. Our data revealed that during mouse and chick development the gene expression pattern changes but does not scale with the overall tissue size. These changes in the pattern are sequentially controlled by distinct mechanisms. Initially, neural progenitors integrate signaling from opposing morphogen gradients to determine their identity by using a mechanism equivalent to maximum likelihood decoding. This strategy allows accurate assignment of position along the patterning axis and can account for the observed precision and shifts of pattern. During the subsequent developmental phase, cell-type specific regulation of differentiation rate, but not proliferation, elaborates the pattern.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Mathematical models of biological systems are often validated by fitting the model to the average of an (often small) experimental dataset. Here we ask the question of whether predictions made from a model fit to the average of a dataset are actually applicable in samples that deviate from the average. We will explore this in the context of a murine model of melanoma treated with oncolytic viruses and dendritic cell injections. We have hierarchically developed a system of ordinary different equations to describe the average of this experimental data, and optimized treatment subject to clinical constraints. Using a virtual population method, we explore the robustness of treatment response to the predicted optimal protocol; that is, we quantify the extent to which the optimal treatment protocol elicits the same qualitative response in virtual populations that deviate from the average. We find that our predicted optimal is not robust and in fact is potentially a dangerous protocol for a fraction of the virtual populations. However, if we consider a different drug dose than used in the experiments, we are able to identify an optimal protocol that elicits a robust anti-tumor response across virtual populations.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

In cancer research most efforts are devoted on the decipher of the IntraTumoral Heterogeneity (ITH). In ITH the action of the evolutionary forces of mutation and selection are essential to determinant the tumor progression, diagnosis and treatment. ITH gives rise to cancer cell populations with distinct genotypic and metabolic characteristics contributing to the failure of cure, by initiating phenotypic diversity and enabling more aggressive and drug resistant clones. I will present multi-scale models of cancer linking the tumor growth to the intracellullar signalling and metabolic events to genomic profiles. The models consider several heterogenous omics data (metabolomics, proteomics, transcriptomics, genomics) to investigate the ITH associated with different genomic and metabolic traits.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Anemia associated with chronic diseases is the second most prevalent anemia in the world after anemia caused by iron deficiency. Advanced stages of diseases such as chronic kidney disease (CKD) and cancer coincide with a high prevalence of severe anemia that results in fatigue, reduced quality of life and decreased treatment responses in patients. Two therapeutic options are available to manage anemia: blood transfusion and treatment with erythropoiesis stimulating agents (ESAs) in combination with iron supplementation. However, adverse events and increased risk of mortality have been reported for blood transfusions and ESAs. Decisions on the clinical treatment should be based on the specific benefit-to-risk ratio of each patient, which is complicated to assess due to the heterogeneity of the patients, the lack of prognostic markers and the dynamics of comorbidities associated with the diseases. We developed a multiscale mathematical model that links mechanistic insights at the cellular scale to response at the body level to guide clinical decisions based on the prediction of the response to the available therapeutic options. The mathematical model stratifies patients based on the estimation of two patient specific dynamic parameters. These parameters are estimated by the mathematical model based on the time-course of the haemoglobin (Hb) values, CRP, iron values and scheduled chemotherapy in each patient. These two patient specific parameters reflect the anaemic status of the patient as well as the capability to respond to treatment with ESAs. The model is capable to propose optimized personalized interventions for anaemia management in lung cancer and CKD patients.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

To tackle the question of drug resistance in cancer, I will present an adaptive dynamic framework to represent the evolution in phenotype of cell populations, that allows to follow the instantaneous distribution and asymptotic behaviour of drug resistance phenotype(s) in the cell population. Such phenotypes evolve under drug pressure towards either established or transient, possibly reversible, drug tolerance, a behaviour taken into account by the models we design to allow for therapeutic control. Optimal control strategies describing the combination of different categories of drugs on specified cell functional targets (thus far cytotoxics, that act on death terms, and cytostatics, that act on proliferation terms) are proposed, aiming at minimising a tumour cell population while limiting both unwanted toxic side effects on healthy cell populations and occurrence of drug resistance in cancer cell populations. The models used for these representations, their asymptotic properties and their theoretical therapeutic control are integro-differential (non-local Lotka-Volterra-like) or PDE models (reaction-diffusion models with or without advection). Finally, I will present some transdisciplinary challenges of cancer modelling that should concern mathematicians, cell biologists, evolutionary biologists and oncologists, aiming to go beyond the present state of the art in the treatments of cancer.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Glioblastoma may have wide-spread effects on the cortical organization and cognitive function since even focal lesions impact the brains’ functional network architecture. Currently, our understanding of the interaction between tumor lesions and their impact on the functional connectome is limited. Hence, we used 3 Tesla resting-state functional magnetic resonance imaging to evaluate the functional connectivity structure of 15 patients with glioblastoma. We further tracked the functional characteristics of six patients over time using bimonthly follow-up examinations. We found changes in resting-state networks to be highly symmetric and mirrored by changes in the cerebellum. Patients shared a pattern of network deterioration after surgery, with subsequent recovery at the first follow-up examination. Additionally, we showed that glioblastoma has a global effect on the functional connectivity structure of the individual patient, which might serve as sensitive early marker of tumor recurrence. Of note, local tumor recurrence coincided with network deterioration before structural changes were apparent upon imaging. In summary, our results demonstrate how the functional connectome is affected by focal lesions, and that it might be exploited as an early predictor of local tumor recurrence. This renders the individual patient’s functional connectome a promising novel biomarker for the longitudinal patient follow-up in order to support early informed treatment decisions.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

In this talk we present our work on the dynamics of tumor and immune cell interactions [1-4]. A hybrid probabilistic cellular automaton model describing the spatio-temporal evolution of tumor growth and its interaction with the cell-mediated immune response is developed. The model parameters are adjusted to an ordinary differential equation model, which has been previously validated [1] with in vivo experiments and chromium release assays. The cellular automaton is used to perform in silico experiments which, together with mathematical analyses, allow us to characterize the rate at which a tumor is lysed by a population of cytotoxic immune cells [2-3]. Finally, the transient and asymptotic dynamics of the cell-mediated immune response to tumor growth is considered [4]. The cellular automaton model is used to investigate and discuss the capacity of the cytotoxic cells to sustain long periods of tumor mass dormancy, as commonly observed in recurrent metastatic disease. This is a joint work with Alvaro G. López and Miguel A. F. Sanjuán.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Cancer imaging has undergone major paradigm shifts within the last decade. Hybrid imaging techniques, and in particular, PET/CT (positron emission tomography / computed tomography) with the glucose analogue radiotracer [18F]FDG is now an integral part of the management guidelines for patients with different cancers, with a particular emphasis on the early detection of treatment effects on the tumor. Novel PET radiotracers that are specific for certain types of cancer – such as [68Ga]PSMA for prostate cancer – are currently being evaluated in clinical trials. Notably, though visual image interpretation is still the clinical standard, there is now a trend towards the use of quantitative data extracted from diagnostic images. The recently introduced PET/MRI (magnetic resonance imaging) is of particular interest in that regard, because it offers information on tissue properties such as cell density and blood flow in addition to the metabolic information provided by PET. The combination of quantitative parameters extracted from MRI and PET may not only improve non-invasive, image-based characterization of tumor heterogeneity, but may also improve evaluation of the effects of novel types of treatment. This multi-parametric approach also provides an ideal basis for radiomics – i.e., computer-assisted image analysis, and based on it, recognition of mathematical image patterns that are related to tumor characteristics. This novel approach to image interpretation, which is aided by advanced techniques such as artificial neural networks, has the potential to contribute significantly to the success of precision medicine, and the welfare of patients.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

In this talk, we investigate the link between multi-scale models for tumor growth. We start from a microscopic model where cells are modelled as 2D spheres undergoing short range repulsion and cell division. We derive the associated macroscopic dynamics leading to a porous media type equation. As the macroscopic equation obtained through usual derivation method fails at providing the correct qualitative behavior, we propose a modified version of the macroscopic equation introducing a density threshold for the repulsion. We numerically validate the new formulation by comparing the solutions of the micro- and macro- dynamics. Moreover, we study the asymptotic behavior of the dynamics as the repulsion between cells becomes singular (leading to non-overlapping constraints in the microscopic model). We show formally that such asymptotic limit leads to a Hele-Shaw type problem for the macroscopic dynamics. The numerical simulations reveal an excellent agreement between the micro- and macro- descriptions, validating the formal derivation of the macroscopic model. The macroscopic model derived here therefore enables to overcome the problem of large computational time raised by the microscopic model, but stays closely linked to the microscopic dynamics.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

In the majority of cancers, secondary tumors (metastases) and associated complications are the main cause of death. To design the best therapy for a given patient, one of the major current challenge is to estimate, at diagnosis, the eventual burden of invisible metastases and the future time of emergence of these, as well as their growth speed. In this talk, I will present the current state of research efforts towards the establishment of a predictive computational tool for this aim. I will first shortly present the model used, which is based on a physiologically-structured partial differential equation for the time dynamics of the population of metastases, combined to a nonlinear mixed-effects model for statistical representation of the parameters’ distribution in the population. Then, I will show results about the descriptive power of the model on data from clinically relevant ortho-surgical animal models of metastasis (breast and kidney tumors). The main part of my talk will further be devoted to the translation of this modeling approach toward the clinical reality. Using clinical imaging data of brain metastasis from non-small cell lung cancer, several biological processes will be investigated to establish a minimal and biologically realistic model able to describe the data. Integration of this model into a biostatistical approach for individualized prediction of the model’s parameters from data only available at diagnosis will also be discussed. Together, these results represent a step forward towards the integration of mathematical modeling as a predictive tool for personalized medicine in oncology.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Oncogenes perturb molecular mechanisms to drive neoplastic initiation and progression. Chromosomal rearrangements are frequent events in cancer, and can result in the expression of fusion proteins. Fusion proteins represent neomorphic protein variants with aberrant activities and are often drivers of oncogenesis. Acute myeloid leukemia (AML) is an aggressive cancer of the white blood cell lineage that is associated with poor prognosis. While AML features a particular high prevalence of fusion proteins, it is largely unknown how the majority of AML fusion proteins rewire the molecular machinery of normal blood cells to induce leukemia. We hypothesize that oncogenic mechanisms of AML fusion proteins are hard-wired in specific networks of physical, genetic and epigenetic interactions with key effector proteins. Functional exploration of these networks by systematic comparative approaches will provide new insights into cellular processes that depend on critical effector proteins among these networks. The goal of our research is a comprehensive systems-level investigation of oncogenic mechanisms employed by AML fusion proteins. We have established a robust experimental pipeline for the rapid characterization of fusion oncoproteins in a multilayered, global fashion. We use modern genetic tools to generate advanced cell and animal models for tunable expression of AML fusion proteins. Fusion protein-dependent changes in cellular topologies are charted by proteomic and transcriptomic approaches. In parallel, genome-scale loss-of function CRISPR/Cas9 screening is used to identify critical effectors of leukemogenesis. High-confidence candidates are validated using a wide array of different approaches, including studies in primary patient-derived leukemia cells. Results from this pipeline provide evidence for its robust validity, but also for its translational impact, strongly implying that this approach will contribute to an improved understanding of oncogenesis.

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

TBA

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

Angiogenesis is the process by which the body generates new blood vessels. This occurs in the context of wound healing where, of course, it is beneficial to the body. However, it can also occur in cancer where it can enhance delivery of nutrients to the cancer and enable cancer cells to infiltrate the blood system and metastasize to vital organs, leading to the often fatal secondary tumours. Understanding this process is a challenge for both experimentalists and theoreticians. I will review some recent work we have done on this problem which includes generating a new partial differential equation model for the so-called ``snail-trail'' movement of blood vessel cells to the tumour (Pillay et al, 2017), by developing a continuuum model of the process from a discrete description. I will then present a computational multiscale model for a key experimental assay that is used by experimentalists to measure the efficacy of anti-angiogenesis drugs and use it to make predictions (Grogan et al, 2018; 2017).

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

• Thematic program: Mathematical models in Biology and Medicine (2018/2019)
• Event: Workshop on "Mathematical Models in Cancer" (2018)

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set $Theta$ of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in $Theta$. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates. Joint work with Tolulope Fadina and Ariel Neufeld.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In a joint work with Walter Schachermayer (still in progress), we investigate the optimal strategy of an economic agent trading a fractional asset in presence of transaction costs. A fascinating conjecture by us asserts that, contrary to the Bronwnian case, such an optimal trading would be fully discrete, only involving countably many trading times. What we can already prove is that only certain specific times, which we call "potential trading times", may involve trading, regardless of the agent's porfolio (this shall be explained more in detail). An idea towards our conjecture (though unsuccessful yet) would be to bound above the fractal dimension of the set of potential trading times. The nice point with this approach is that, contrary to the optimal strategy, this fractal dimension can be computed numerically: the goal of my talk will be to explain how one can do so. The method I propose involves quasi-stationary distributions, that is, killed Markov processes conditioned by long-time survival: which is rather surprising, as this concept has a priori nothing to do with fractal dimension ...

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classica affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. Nonetheless, their Fourier-Laplace functionals admit exponential-affine representations in terms of solutions of associated deterministic integral equations, extending the well-known Riccati equations for classical affine diffusions. Our findings generalize and clarify recent results in the literature on rough volatility.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

We introduce a new method to price American options based on Chebyshev interpolation. The key advantage of this approach is that it allows to shift the model-dependent computations into an offline phase prior to the time-stepping. This leads to a highly efficient online phase. The model-dependent part can be solved with any computational method such as solving a PDE, using Fourier integration or Monte Carlo simulation.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

We will discuss non-exponential versions of well known limit theorems, specialising on the case of Brownian motion. The proofs will partially rely on the theory of BSDEs and their convex dual formulations, and an application to (stochastic) optimal transport will be provided.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

When we are ill, most of us would prefer to receive treatment that was supported by scientific evidence, rather than anecdotal tradition or superstition. When a nation's economy is ill, policy-makers turn to economists for advice, but how well is their advice supported by evidence? This talk critiques the value of economic theory in practice, and tries to suggest ways of increasing the practical relevance of the subject.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are also highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In this talk we introduce of a class of new stochastic approximation algorithms for high-dimensional nonlinear PDEs. We prove that these algorithms do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solutiothe approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

A market of financial securities with restricted participation is considered. Agents are heterogeneous in beliefs, risk tolerance and endowments, and may not have access to the trade of all securities. The market is assumed thin: agents may influence the market and strategically trade against their price impacts. Existence and uniqueness of the equilibrium is shown, and an efficient algorithm is provided to numerically obtain the equilibrium prices and allocations given market’s inputs. (Based on joint work with M. Anthropelos.)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Probability-valued jump-diffusions provide useful approximations of large stochastic systems in finance, such as large sets of equity returns, or particle systems with mean-field interaction. The dynamics of a probability-valued jump-diffusion is governed by an integro-differential operator of Levy type, expressed using a notion of derivative that is well-known from the superprocesses literature. General and easy-to-use existence criteria for probability-valued jump-diffusions are derived using new optimality conditions for functions of probability arguments. In general, we consider the space of probability measures as endowed with the topology of weak convergence. For jump-diffusions taking value on a specific subset of the probability measures, it can however be useful to work with a stronger notion of convergence. Think for instance at the well-known Wasserstein spaces. This change of topology permits to include in the theory a larger class of generators, and hence, a larger class of probability-valued jump-diffusions. We derive general and easy-to-use existence criteria for jump-diffusions valued in those spaces.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In this talk I will explain what is a Variable Annuities (VA) contract and how we can find the pricing formula of VA when the financial market is hybrid in the sense introduced by Eberlein.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

TBA

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Insurance contracts are efficient risk management techniques to operate and reduce losses. However, very often, the underlying probability model for losses - on the basis of which premium is computed - is not completely known. Furthermore, in the case of extreme climatic events, the lack of data increases the epistemic uncertainty of the model. In this talk we propose a method to incorporate ambiguity into the design of an optimal insurance contract. Due to coverage limitations in this market, we focus on the limited stop-loss contract, given by $I(x)=min(max(x-d_1),d_2)$, with deductible $d_1$ and cap $d_2$. Therefore, we formulate an optimization problem for finding the optimal balance between the contract parameters that minimize some risk functional of the final wealth. To compensate for possible model misspecification, the optimal decision is taken with respect to a set of non-parametric models. The ambiguity set is built using a modified version of the well-known Wasserstein distance, which results to be more sensitive to deviations in the tail of distributions. The optimization problem is solved using a distributionally robust optimization setup. We examine the dependence of the objective function as well as the deductible and cap levels of the insurance contract on the tolerance level change. Numerical simulations illustrate the procedure.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In this talk, we propose a novel approach to the modelling of multiple yield curves. Adopting the HJM philosophy, we model term structures of forward rate agreements (FRA) and OIS bonds. Our approach embeds most of the existing approaches and additionally allows for stochastic discontinuities. In particular, this last feature has an important motivation in interest rate markets, which are affected by political events and decisions occurring at predictable times. We study absence of arbitrage using results from the recent literature on large financial markets and discuss special cases and examples. This talk is based on joint work with Zorana Grbac, Sandrine Gümbel and Thorsten Schmidt.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

I'll try in this talk to present the main ideas on zero-sum continuous time games where one of the two players has some private information (for instance when only one player observes a Brownian motion): how to formalize these games, the associated Hamilton-Jacobi-Isaacs-equation and the analyse of the optimal revelation in terms of an optimization problem over a set of martingales. In a second time I'll present the last developments in this area.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Logic, Data, Advanced Information systems (2017/2018)
• Event: Workshop on "Deontic Reasoning: From Ancient Texts to Artificial Intelligence" (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: Workshop on “Numerics for cosmology – the Schrodinger method” (2018)

A holepunch cloud is a curious and rare atmospheric feature where an aircraft, descending or ascending through a thin cloud layer, leaves behind a growing circular hole of clear air. Observed since the early days of aviation, only in 2011 was this holepunch phenomenon simulated in a full-physics numerical weather model. Although the initiation process has long been explained by ice crystal formation, the continued growth of the hole, even up to an hour after its birth, remained a bit of a fluid dynamical mystery. We begin by excluding some of the ``obvious" reasons by tweaking the physics in the numerical simulations (fake weather!). We then attribute the expansion of the hole to the presence of an expanding wavefront. The leading edge of this wave is a front of phase change, where cloudy air is continually evaporated and so expands the hole. Our explanation has led us towards the development of a more general theory for an understanding of how atmospheric waves can evolve the shape of clouds. This work is in collaboration with R Rotunno (NCAR), H Morrison (NCAR), R Walsh (SFU) and H Lynn (SFU).

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)

In this talk, we will present some recent results on decay to zero for linear kinetic models with weak or without space confinement. Joint with Mouhot, Mischler, Dolbeault, Schmeiser.

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)

Motivated by recent papers describing rules for natural network formation in discrete settings, we propose an elliptic-parabolic system of partial differential equations. The model describes the pressure field due to Darcy’s type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate representing randomness in the material structure. After a short overview of the principles of discrete network modeling, we show how to derive the corresponding macroscopic (continuum) description. The highly unusual structure of the resulting PDE system induces several interesting challenges for its mathematical analysis. We give a short overview of the tools and tricks that can be used to overcome them. In particular, we present results regarding the existence of weak solutions of the system, based on recent results on elliptic regularity theory. Moreover, we study the structure and stability properties of steady states that play a central role to understand the pattern capacity of the system. We present results of systematic numerical simulations of the system that provide further insights into the properties of the network-type solutions.

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)

We consider a regularisation of a scalar conservation law where the viscous term is a Caputo type fractional derivative of order between 1 and 2. We shall first focus on some recent results on the study travelling wave solutions of the Korteweg-de Vries-Burgers equation with such non-local viscous term, the third order one being local and linear. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). We show rigorously the existence of these waves in the case of a genuinely non-linear flux and for the case of a non genuinely non-linear one, we give results on the existence of the waves that do not satisfy the entropy condition. We shall also discuss the vanishing viscosity limit when the third order term is not present.

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)

We are interested in evolutionary biology models for sexual populations. The sexual reproductions are modelled through the so-called Infinitesimal Model, which is similar to an inelastic Boltzmann operator. This kinetic operator is then combined to selection and spatial dispersion operators. In this talk, we will show how the Wasserstein estimates that appear naturally for the kinetic operator can be combined to estimates on the other operators to study the qualitative properties of the solutions. In particular, this approach allows us to recover a well-known (in populations genetics) macroscopic model.

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)

We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations: the tools developed for studying elliptic or parabolic equations with rough coefficients on the one hand (De Giorgi, Nash, Moser, Krylov, Safonov), and the theory of hypoellipticity (H\"ormander) on the other hand. We discuss recent results about hypoelliptic equations of kinetic type with rough coefficients. We then discuss applications to the Boltzmann and Landau equations and present a program of research about the regularity for these equations, with some open questions.

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)

Entropy-based methods, and in particular the so-called "generalised relative entropy" inequalities, have been developed and successfully applied to structured population equations, and in particular to aggregation-fragmentation problems, over the last two decades. In this talk, we study how entropy methods have been recently extended to measure solutions [1] as well as to the convergence towards a periodic limit [2]. We also investigate the long-time dynamics of a family of nonlinear nucleation-aggregation equations, for which specific entropy functionals may be built [3]. Ref: [1] Thomasz Debiec, Marie Doumic, Piotr Gwizada, Emil Wiedemann, Relative entropy method for measure solutions of a structured population model, 2018 [2] Etienne Bernard, Marie Doumic, Pierre Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, 2016 [3] Juan Calvo, Marie Doumic, Benot Perthame, Long-time asymptotics for polymerization models, 2017

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Applied PDEs and kinetic equations: from physics to life sciences and beyond" (2018)