Abstracts of talks

Szymon Brzostowski (Uniwersytet Łódzki)

The Łojasiewicz exponent and polar curves

Abstract. Let \(f:(\mathbb{C}^n,0)\to(\mathbb{C},0)\) be a holomorphic function with an isolated singular point at the origin. B. Teissier proved that the Łojasiewicz exponent \(ł(f)\) of \(f\) is attained on generic polar curves of \(f\). In this talk, we discuss the effects of focusing on (much simpler in practice) ''coordinate polar curves'' for the computation of the Łojasiewicz exponent.

Paweł Goldstein (Uniwersytet Warszawski)

Constructing homeomorphisms and diffeomorphisms with prescribed derivatives

Abstract. The problem of reconstructing a mapping from its derivative \(f\) is as old as calculus. Particularly, understanding the necessary conditions for f to be a derivative (a gradient, a derivative a.e. etc) proved to be particularly fruitful for mathematics. In my recent joint work with Zofia Grochulska and Piotr Hajłasz, we addressed a particular instance of the problem: what are the necessary conditions for a measurable \(F:[0,1]^n\to GL(n)\) to be a derivative of a (suitably differentiable) homeomorphism? We obtained a rather surprising result: essentially none. For any such \(F\) we construct an a.e. approximately differentiable* homeomorphism \(H\) of a cube with \(DH=F\) almost everywhere. As a by-product, we obtain a related result for diffeomorphisms: if only \(det F>0\) a.e., then there is a diffemorphism \(H\) of a cube with \(DH=F\) outside a set of arbitrarily small measure.

* a very natural class in geometric measure theory

Rouzbeh Mohseni (IMPAN)

Sasakian manifolds: Old and New

Abstract. Sasakian manifolds were introduced by Sasaki in 1960, and in a sense, they are the odd-dimensional counterpart of Kähler manifolds. In this talk, I will first review several approaches used to define the Sasakian manifolds and then discuss new ways with examples of obtaining them.

Tadeusz Mostowski (Uniwersytet Warszawski)

On Marie-Hélène Schwartz's construction of Chern classes in the Lipschitz framework

Abstract. Chern classes were constructed by Marie-Hélène Schwartz using the notion of obstruction theory. Her work preceded the paper of MacPherson. Her approach was difficult because of some problems in singularity theory. In this talk, I will explain how to eliminate these problems using Lipschitz stratifications.

Rafał Pierzchała (Uniwersytet Jagielloński)

On the \(\mathcal{C}^p_\omega\) category

Abstract. I will present some new results concerning the class \(\mathcal{C}_\omega^p\), where \(\omega\) is a modulus of continuity.

Artur Piękosz (Politechnika Krakowska)

The development of tame topology

Abstract. A. Grothendieck postulated the development of tame topology. We concentrate on the purely topological aspects of this search. Starting with Grothendieck sites and Gtopologies, we show how the notion of a tame space was concretized. We demonstrate how some fundamental dualities in topology can be applied to the tame context, producing many equivalences and concrete isomorphisms. Looking from the o-minimal perspective, we explore the possibilities of applying our approach to a more general model-theoretical setting.

Anna Valette (Uniwersytet Jagielloński)

Semialgebraic Whitney partition of unity

Abstract. I will present a notion of \(\Lambda_p\)-regular partition of unity which can be seen as a semialgebraic counterpart of Whitney partition of unity. This enables us to obtain a semialgebraic (or more generally definable) version of Calderón-Zygmund theorem on regularization of the distance function. Some more consequences will also be  given.  This  is based on a common work with Wiesław Pawłucki and Beata Kocel-Cynk. 

Andrzej Weber (Uniwersytet Warszawski)

How to arrive from nothing to elliptic genus?

Abstract. Many invariants of singular spaces are computed using resolutions of singularities. Small resolutions, which are resolutions with an exceptional locus of codimension at least two, play a special role. However, small resolutions do not always exist, and when they do, they might be not unique. Similar issues arise for symplectic resolutions in the sense of Beauville. Therefore, to define an invariant of a singular variety, one must:

1. Fix an invariant for smooth varieties.

2. Ensure that it remains unchanged under modifications of small resolutions or symplectic resolutions.

We will focus on characteristic classes obtained through the application of the Hirzebruch formalism and demonstrate that only the elliptic class and its degenerations satisfy condition (2). Consequently, we recover a result of Burt Totaro and extend it to symplectic varieties.