### Research seminar

Organized jointly with A. Kriegl and P. Michor-
Rafal Pierzchala (Jagiellonian University)

*Remez-type estimates*

Abstract

May 4, 2023, 13:30 Seminarraum 06 Oskar-Morgenstern-Platz 1 1.Stock -
M'hammed Oudrane (Université Côte d'Azur)

*Sobolev sheaves on the subanalytic topology*

Sheaves on manifolds are good objects to deal with local problems, but from the point of view of algebraic geometry, the usual topology contains many open sets of pathological nature, which makes the family of open subanalytic sets (or definable sets in some fixed o-minimal structure) a good candidate for replacing the usual topology. On the subanalytic topology, sheaves that are defined by functional spaces are very important in the study of irregular holonomic D-modules, but unfortunately many functional spaces are not of local nature. In this talk, we present G.Lebeau's method of sheafying (in the derived sense) the Sobolev spaces H^{s}on the subanalytic topology for s ≤ 0, and we present a method to construct these sheaves (in the usal sense) for s ≥ 0 in dimension 2, based on the geometric nature of open subanalytic sets in R^{2}. We give also a possible construction for the higher dimensional case.

April 5, 2022, 10:00 Seminarraum 07 Oskar-Morgenstern-Platz 1 2.Stock -
Adam Parusinski (Université Côte d'Azur)

*Singularities of Algebraic Hypersurfaces in Codimension 2*

In 1979 O. Zariski proposed a general theory of equisingularity for algebraic or algebroid hypersurfaces over an algebraically closed field of characteristic zero. This theory is based on the notion of dimensionality type that is defined recursively by considering the discriminants loci of subsequent "generic" projections. Thus the points of dimensionality type 0 are regular points and the singularities of dimensionality type 1, are generic singular points in codimension 1. Zariski proved that the latter ones are isomorphic to the equisingular families of plane curve singularities. In this talk we give a similar characterization for singularities of dimensionality type 2, i.e. for generic singularities in codimension two. We show that they are isomorphic to equisingular families of surface singularities, with the equisingularity type determined by the discriminants of their "generic" projection. Moreover, we show that in this case the generic linear projections are generic (this is still open for dimensionality type greater than 2). Over the field of complex numbers, we show that such families are bi-Lipschitz trivial, by constructing an explicit Lipschitz stratification. (Based on joint work with L. Paunescu.)

February 27, 2020, 14:00 Seminarraum 08 Oskar-Morgenstern-Platz 1 2.Stock -
Vicente Asensio (Universitat Politecnica de Valencia)

*Global pseudodifferential operators in classes of ultradifferentiable functions and applications*

In this seminar, we deal with pseudodifferential operators of infinite order in classes in the spirit of Björck for non-quasianalytic weight functions in the sense of Braun, Meise and Taylor. We also discuss sufficient conditions for the construction of a parametrix and we apply those results to the regularity of partial differential operators. This is based on collaborations with Chiara Boiti, David Jornet and Alessandro Oliaro.

February 20, 2020, 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock -
Peter Michor (Universität Wien)

*General Sobolev metrics on the manifold of all Riemannian metrics: Smoothness of the fractional Laplacian on the space of Riemannian metrics of Sobolev order*

For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}_{H^s}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s<\frac m2$ with respect to $\hat g$. The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order $s<\frac m2$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. The finally complete proof of well-posedness involves complex interpolation spaces and, sectorial operators, and bounded $\mathcal H^\infty$ -calculus.

October 22, 2018, 13:30 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock -
Cornelia Vizman (West University of Timisoara)

*Transgression of differential characters to spaces of functions / submanifolds.*

Joint work with Tobias Diez, Karl-Hermann Neeb, and Bas Janssens. Differential characters of degree one are in bijection with isomorphism classes of principal circle bundles with connection, via the holonomy map. We define differential characters of higher degree (higher dimensional holonomy) and we describe some of their properties following [BB]. For a compact manifold $S$, we show how differential characters on $C^\infty(S,M)$, as well as on the nonlinear Grassmannian $Gr^S(M)$ of submanifolds of $M$ of type $S$, are obtained by combining in a natural way differential characters on S and on M. The aim is to obtain degree one differential characters on these Fr\'echet manifolds, in order to use the prequantization central extension for integrating Lichnerowicz 2-cocycles on the Lie algebra of divergence free vector fields.

[BB] Christian Baer and Christian Becker, Differential Characters, Lecture Notes in Mathematics 2112, Springer 2014.

June 27, 2018, 13:15, Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock -
Sergio Carrillo (Universität Wien)

*Tauberian theorems for summability in analytic functions.*

(Joint work with R. Schäfke and J. Mozo) In a recent work, R. Schäfke and J. Mozo introduced the notions of asymptotic expansions and summability with respect to a germ of an analytic function $P$ in several variables. The goal of this talk is to generalize the tauberian properties for k-Borel-summability to this new setting. In particular we prove that a series $P_0-k_0-$ and $P_1-k_1-$summable is convergent, unless there are positive integers $p_0, p_1$ and a unit $U$ such that $p_0/k_0=p_1/k_1$ and $P_1^{p_1}= U P_0^{p_0}$, in which case the summability processes are the same.

June 7, 2018, 13:30, Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock -
Tobias Kaiser (Universität Passau)

*Hardy fields, o-minimal structures, and connections to Hilbert 16.*

In joint work with Patrick Speissegger we have recently constructed a Hardy field which contains all transition maps of plane polynomial vector fields at hyperbolic singularities and all unary functions definable in the o-minimal expansion of the real field by restricted analytic functions and exponentiation. In this talk I explain these notions and their connections and discuss how o-minimality can make some contributions to questions around Hilbert 16.

May 17, 2018, 13:30, Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock -
Jorge Mozo (Universidad de Valladolid)

*Results about analytic classification of holomorphic foliations - dimension three.*

We shall first review the main known results concerning the analytic classification of germs of codimension one singular holomorphic foliations, in dimension two and three. We focus in the works from Cerveau, Moussu, Meziani, Berthier, Sad, Strozyna and others, in the nilpotent case. The state-of-art of this subject in dimension three will be explained. In particular, how the existence of a (pre)-normal form may be helpful, and which are some of the techniques involved. This is included in several works in collaboration with Percy Fernández and Hernán Neciosup (LIma, Perú).

May 9, 2018, 13:30, Seminarraum 13 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*Tauberian theorems for polynomial summability.*

(Joint work with R. Schäfke) The aim of this talk is to present comparison results between methods of summability defined through monomial and polymomial asymptotic expansions in several variables. As a main result we prove that two germs of analytic maps define the same 1-summability process if and only if they differ by a unit. The exposition is based on the well-known results in one variable for Borel k-summability and the process of monomialization of germs of analytic functions.

April 26, 2018, 13:30, Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock -
Reinhard Schäfke (Université de Strasbourg)

*Asymptotic expansions with respect to an analytic germ.*

(Work in common with Jorge Mozo-Fernández) In a previous article by the authors, monomial asymptotic expansions, Gevrey asymptotic expansions and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables - this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis-Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation.

April 19, 2018, 13:30, Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock -
Adam Parusinski (Université de Nice)

*Sobolev sheaves on subanalytic sites.*

This talk is meant to be a non-technical introduction to a recent Astérisque entitled "Subanalytic sheaves and Sobolev spaces". The motivation comes from the sheaf of tempered distributions introduced by Kashiwara. The linear subanalytic sites of Guillermou-Schapira allows one to treat the more sophisticated function spaces such as the Gevrey functions with given order, or the Sobolev spaces. Sobolev sheaves are the (complexes of) sheaves on subanalytic sites that coincide with the classical Sobolev spaces on the subanalytic Lipschitz domains.

March 22, 2018, 13:30, Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock -
Peter Michor (Universität Wien)

*Soliton solutions for the elastic metric on spaces of curves.*

The lecture starts with a general discussion about the nature of solitons. Then a special case will be discussed in detail: Some first order Sobolev metrics on spaces of curves admit soliton-like geodesics, i.e., geodesics whose momenta are sums of delta distributions. It turns out that these geodesics can be found within the submanifold of piecewise linear curves, which is totally geodesic for these metrics. Consequently, the geodesic equation reduces to a finite-dimensional ordinary differential equation for a dense set of initial conditions.

January 25, 2018, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*On asymptotic expansions in several variables and applications to singularly perturbed first order PDEs.*

The goal of this talk is to describe some of the relations and properties between different approaches to asymptotic expansions for formal power series in several variables such as monomial and Mayima's expansions. The Borel-Laplace analysis for each case is also described as a tool to prove monomial summability of solutions of singularly perturbed PDEs of the form $$x^\alpha\varepsilon^\beta\sum_{j=1}^n s_j/\alpha_j x_j \partial y/\partial x_j =F(x,\varepsilon,y),$$ where $x=(x_1,...,x_n), \varepsilon=(\varepsilon_1,...,\varepsilon_m)$ are tuples of complex variables, $\alpha=(\alpha_1,...,\alpha_n), \beta=(\beta_1,...,\beta_m)$ are tuples of positive integers, $s_1,...,s_n$ are non-negative real numbers such that $s_1+...+s_n=1$, $F$ is analytic in a neighborhood of the origin and $\partial F/\partial y(0,0,0)$ is an invertible matrix. The results presented here are a first step on summability of formal solutions that are expected to be valid for more general equations of the same nature.

January 18, 2018, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*A link between Denjoy-Carleman classes on sectors in a monomial and monomial asymptotic expansions.*

The aim of this talk is to prove that it is equivalent for a function to have asymptotic expansion w.r.t. a monomial and having bounded derivatives of all orders in every monomial subsector of its domain. We will explain the analogous result for the case when the asymptotic or the growth of the derivatives is determined by a weight sequence, logarithmically convex and stable by derivatives.

November 16, 2017, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*Spaces of asymptotically developable functions III. An extension to several variables through monomials.*

The goal of this talk is to extend the concept of asymptotic expansions and k-summability to several variables using parametrizations given by monomials. We will explore the main properties of such notions including their behaviour under point blow-ups and its applications to some singularly perturbed differential equations.

October 19, 2017, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*Spaces of asymptotically developable functions II.*

The aim of this talk is to continue the study of k-summability and its applications. As a main example we will study the Heat equation in the holomorphic setting and we will establish conditions on the initial condition to conclude summability properties of the solutions. In particular we show how to recover the Heat Kernel through the Stokes phenomenon for a suitable initial condition. We will also include an initial approach to asymptotic and summability in several variables.

October 12, 2017, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Sergio Carrillo (Universität Wien)

*Spaces of asymptotically developable functions I. The case of one variable.*

The goal of this talk is to introduce the notion of asymptotic expansions and Borel k-summability in the case of one variable. We will explore the main tools in this theory including Ramis-Sibuya theorem and the use of Borel and Laplace integral operators to compute sums. An application of summability of formal power series solutions of analytic ODEs of first order at irregular singular points will be given.

October 5, 2017, 13:45, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Vincent Grandjean (Universidade Federal do Ceara, Fortaleza)

*Geodesics at singular point: on the problem of the exponential map*

Any Riemannian manifold M admits at each point a neighbourhood over which exist polar-like coordinates, namely normal coordinates. Assuming given a subset X of M which is not submanifold, we can nevertheless equip its smooth part with the restriction of the ambient Riemmannian structure and try to understand the behaviour of geodesics nearby any non smooth point. The most expected occurrence of such situation is when M is an affine or projective space (real or complex) and X is an affine or projective variety with non-empty singular locus. The standard strategy is to use a parameterization of X (resolution of singularities) in such a way that the source space is a manifold with boundary (mapped surjectively onto the singular locus) and pull back the Riemannian structure onto this manifold, using this parameterization, and work nearby the boundary with a degenerate tensor along the boundary. In a joint work with D. Grieser (Univ. Oldenburg, Germany) we discuss the problem of an exponential-like map at the singular point of a class of isolated surface singularities of an Euclidean space, called cuspidal surface, which are explicit in some sense. I will state the trichotomy of this class of surface regarding the existence and the injectivity of an exponential-like mapping at the singular point of this class of surface... and explain a bit if times allows.

January 26, 2017, 15:00, Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock -
Cornelia Vizman (West University of Timisoara)

*Central extensions of Lie algebras of symplectic and divergence free vector fields*

We present the classification of continuous central extensions of the Lie algebra of Hamiltonian vector fields, together with its universal central extension, comparing the results with those for the Lie algebra of divergence free vector fields. We discuss also integrability issues. This is joint work with Bas Janssens from Utrecht University

June 2, 2016, 15:30, Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock -
Peter Michor (University of Vienna)

*Stokes' and Moser's theorem for manifolds with corners*

Moser's theorem 1965 states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners which is essentially due to Banyaga 1974. Simplices are a particular case. A cohomological interpretation of Banyaga's operator is given which allows a differential form proof of Lefschetz duality.

April 14, 2016, 15:30, Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock -
Martins Bruveris (Brunel University London)

*Regularity of the geodesic boundary value problem on the diffeomorphism group*

I will talk about a result that is inspired from and a generalization of results on the smoothness of geodesics for right-invariant Riemannian metrics on the diffeomorphism group. Riemannian metrics on the diffeomorphism group are being used in shape analysis to to drive deformations of images. Under some natural assumptions, a right-invariant metric gives rise to a smooth, right-invariant exponential map on the group $D^q(R^d)$ of Sobolev diffeomorphisms with $q$ large enough. The right invariance leads to the following property: if the initial conditions of a geodesic are of class $H^{q+k}$, then so is the whole geodesic. This implies that smooth initial conditions lead to smooth geodesics. In this talk I will show how to generalize this regularity principle to show the corresponding statement about the boundary value problem: if two diffeomorphisms and are nonconjugate along a geodesic, then the geodesic is as smooth as the boundary points. This result also holds on diffeomorphism groups of compact manifolds and spaces of curves and surfaces.

April 7, 2016, 15:30, Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock -
Adam Parusinski (Universite de Nice)

*Local topological algebraicity of analytic function germs*

We show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety. The main tools for the proof are: Artin approximation and Zariski equisingularity.

March 17, 2016, 15:30, Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock