Research seminar
Organized jointly with A. Kriegl and P. Michor
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Rafal Pierzchala (Jagiellonian University)
Remez-type estimates
Abstract
May 4, 2023, 13:30
Seminarraum 06 Oskar-Morgenstern-Platz 1 1.Stock
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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 Hs 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 R2.
We give also a possible construction for the higher dimensional case.
April 5, 2022, 10:00
Seminarraum 07 Oskar-Morgenstern-Platz 1 2.Stock
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
