School on Singularity Theory (15 - 21 July 2018)

Minicourses: There will be four minicurses. See timetable below.

Title: Characteristic classes of singular varieties
Lecturer: Prof. Jörg Schürmann (Westfälische Wilhelms-Universität Münster, Germany)
Abstract: First we introduce the calculus of constructible functions for Whitney stratified spaces in the context of real and complex geometry. Next we will explain the isomorphism between constructible functions and Lagrangian cycles (in an embedded context), using the language of "stratified Morse theory for constructible functions". Then we discuss applications to Poincare-Hopf theorems as well as the Lagrangian approach to Stiefel-Whitney- resp. Chern-classes for singular spaces in the context of real resp. complex geometry. Finally we consider more recent results like equivariant versions or alternatively the motivic Hirzebruch- and Chern-classes, maybe with illustrations for Schubert or toric varieties.
References:
1) Schürmann, Jörg: "Lectures on characteristic classes of constructible functions." Notes by Piotr Pragacz and Andrzej Weber. Trends Math., Topics in cohomological studies of algebraic varieties, 175–201, Birkhäuser, Basel, 2005.
2) Schürmann, Jörg: "Chern classes and transversality for singular spaces." Singularities in geometry, topology, foliations and dynamics, 207–231, Trends Math., Birkhäuser/Springer
3) Brasselet, Jean-Paul; Schürmann, Jörg; Yokura, Shoji: "Hirzebruch classes and motivic Chern classes for singular spaces." J. Topol. Anal. 2 (2010), no. 1, 1–55.

Title: Topology of real algebraic sets
Lecturer: Prof. Selman Akbulut (Michigan State University , USA)
Abstract: This will be a mini-course on real algebraic sets. We will discuss the topological characterization problem of real algebraic sets, i.e. “which topological spaces can be described as real algebraic sets?” As time permits after an elementary introduction to real algebraic sets, starting with the case of smooth manifolds, and PL manifolds, we will move the discussion to the stratified sets with “topological resolution tower" (as topological models for real algebraic sets) which give combinatorial invariants, as obstruction a space to be homeomorphic to a real algebraic set. Along the way, we will discuss the notions of "algebraic homology" and "transcendental smooth manifolds", and mention some open problems. All these topics are from my long time joint work with Henry C. King.

Title: Computational Methods for calculating multiple point spaces of map germs and applications (in Portuguese)
Lecturer: Prof. Aldicio José Miranda (Universidade Federal de Uberlândia, Brazil)
Abstract: The multiple point spaces of a map germ f play an important role in the study of its geometry, as well as the topology of the image or discriminant of a stable perturbation of f. I will introduce some algorithms and implementations on Maple and Singular to obtain the definition ideals of such multiple point spaces in the source and in the target. The aim of the mini-course is to introduce the students and researchers to the use of computational methods for studying properties of explicit examples of singularities.
Course notes: click here to download

Title: O teorema dos quatro vértices e sua recíproca (in Portuguese)
Lecturer: Prof. Ronaldo Garcia (Universidade Federal de Goiás, Brazil)
Abstract: Temos como objetivo desenvolver conceitos e ferramentas de geometria diferencial de curvas planas para expor várias provas conceitualmente diferentes do Teorema dos Quatro Vértices (uma curva plana, regular, simples, fechada tem pelo menos quatro vértices) e uma recíproca. Generalizações, assuntos relacionados e conceitos da Teoria de Singularidades serão enfatizados.
Course notes: click here to download