Abstract (PDF file 75kb, updated 12 Feb 2024)                                  “***” indicates an online lecture.

Monday 4th March

Title: On stratified spaces

By:   Shoji Yokura

         (Kagoshima University)

Abstract:

A stratified space is a topological space equipped with what is called stratification, which is a decomposition or partition of the space into disjoint subspaces called strata, satisfying some reasonable conditions. Another one is a poset-stratified space, which is a continuous function from a topological space to the Alexandrov space associated to a poset, a partially ordered set. In this talk I will discuss about relations between stratification and a poset-stratified space.  If time permits, I will talk about some related topics as well.

Title: A conceptual introduction to Schwartz distributions and Colombeau generalized functions

By:    Paolo Giordano

         (University of Vienna)

Abstract:

The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. But also within mathematics, L. Hörmander stated: ``In differential calculus one encounters immediately the unpleasant fact that not every function is differentiable.  The purpose of distribution theory is to remedy this flaw; indeed, the space of distributions is essentially the smallest extension of the space of continuous functions where differentiability is always well defined''.   We first describe the universal property of the space of distributions, but then we underscore the main deficiencies of this theory: we cannot evaluate a distribution at a point, we cannot make non-linear operations, let alone composition, we do not have a good integration theory, etc.  We then present Colombeau theory of generalized functions, which is able to overcome several drawbacks of Schwartz distributions: pointwise evaluation, non-linear operations, partially defined composition, etc. Finally, we explain what are the limitations of Colombeau theory and where they originate from.  This first talk aims to introduce the most well-known theories where continuous functions share several properties with smooth ones, a key step to build-up differential homotopy theory.

Title: Nonstandard diffeology and generalized functions

By:    Kazuhisa Shimakawa

         (Okayama University)

Abstract:

In this talk I introduce a nonstandard extension (in the sense of A. Robinson) of the notion of diffeological spaces, and demonstrate its application to the theory of generalized functions.  Just as diffeological spaces can be defined as concrete sheaves on the site consisting of Euclidean open sets and smooth maps, nonstandard diffeological spaces are defined as concrete sheaves on the site consisting of open subsets of nonstandard Euclidean spaces.  By utilizing this similarity, we can show that nonstandard diffeological spaces form a super-category of the category of diffeological spaces which is self-enriched,

complete, cocomplete and cartesian closed.

  As a first application of our category, we show that the space of nonstandard functions is a smooth differential algebra over (a variant of) Robinson's field of nonstandard numbers and there is a linear injection of the differential vector space of Schwartz distributions into the differential algebra of nonstandard functions.  In this regard, our algebra of nonstandard functions plays a role similar to Colombeau's algebra.  But our algebra has an advantage over Colombeau's one in that it enables not only the multiplication of distributions but also the composition of them because it is a hom-object in a category.

  P. Giordano has already constructed similar differential algebra by extending Colombeau's construction.  Among other differences, the scalar of our theory is not Colombeau's ring, which is a ring with zero-divisors, but a (non-Archimedean) real closed field introduced by A. Robinson.  This of course means that Colombeau's algebra cannot be embedded into our algebra.  Still, it can be shown that there is a chain of algebra homomorphisms connecting the two algebras.

Title: Construction of string homology of submanifolds by de Rham chains

By:    Yukihiro Okamoto

         (Kyoto University)

Abstract:

We fix a smooth manifold.  For any compact smooth submanifold of codimension 2, Cieliebak-Ekholm-Latschev-Ng defined a chain complex whose homology they called the string homology.  The construction involves a chain-level coproduct operation inspired by string topology, but it is restricted in lower degrees.  They also proved that when the submanifold is a knot in R^3, the zeroth degree part of the string homology is isomorphic to an algebraic invariant which derives from Floer theory in contact topology.  In this talk, I will explain how to extend the string homology for submanifolds of arbitrary codimension, though the coefficient is reduced from the original one.  The key point is that we use de Rham chains (instead of singular chains) of differentiable spaces of paths defined by Irie because they are suitable to define the chain-level operation in higher degrees.  I will also show some examples of computations.  If time permits, I will explain a perspective of connecting the string homology to Floer theory in contact topology.

Title: On the geometry of equation manifoldss ***

By:    Jean-Pierre Magnot

         (University of Angers)

Abstract:

The notion of equation manifold is an underlying topic in the theory of differential equations. As subsets of jet spaces, equation manifolds carry interesting geometric properties linked with geometric invariants of the underlying differential equation. In this talk, we will see how diffeologies can encompass the topological pathologies that one experiences, in particular when working on a non compact base space, then we will express globally some geometric properties that are only described on loci on a fixed equation manifold in the existing literature. We will finish with concerns on symmetries, currents, deformations, and the (diffeological) relationship between a wide class of equation manifolds with algebraic curves. This talk is based on a research program in progress, and we will try to include open directions in the exposition of the results recently published or pre-published.