MINICURSOS

Camilo Arias, Universidad Nacional de Colombia, Sede Medellín
Título: The heat equation, Chern Weil theory and the Index of Dirac Operators
Resumen: We begin by introducing the heat equation, exploring its discrete formulation as motivation, describing explicit solutions, and the heat kernel method. Next, we will turn to Chern-Weil theory, which is the geometric construction of the characteristic classes that provide topological invariants of principal bundles. We will then introduce Dirac operators, which generalize the operator introduced by Paul Dirac in the study of a relativistic theory of the electron. Finally, we will describe how the heat kernel method can be used to provide a proof of the Atiyah-Singer Index theorem for Dirac operators. As special cases we will discuss the Chern-Gauss-Bonnet theorem and Hirzebruch's signature theorem. The goal of the course is to highlight the interplay between probability theory, geometry, topology and analysis.

Julie Bergner, University of Virginia
Título: Homotopy-theoretic approaches to categories
Resumen: The goal of this mini-course is to introduce some of the approaches to $(\infty,1)$-categories that are commonly used in homotopy theory and related fields. We'll start with some necessary introductory category theory, then introduce simplicial sets and more general simplicial objects. Finally, we'll look at the motivation for and definitions of some of the different models for $(\infty,1)$-categories.

Mike Hill, University of California, Los Angeles
Título: Equivariant Homotopy Theory
Resumen: Ordinary homotopy theory studies topological spaces “up to homotopy”, describing algebraic invariants like homotopy and homology groups which depend only on homotopy classes of maps. Equivariant homotopy does something similar, but here, we work with spaces together with an action of a fixed group. The goal of the mini-course will be to explain the basic features of this, working towards understanding how to compute with and work with these objects.

Fabiola Manjarrez, IM-UNAM Cuernavaca
Título: Invariants of manifolds and handle decompositions
Resumen: This course focuses on manifolds of dimensions 1, 2 and 3. All of them can be decomposed into handles of their respective dimensions. Such decompositions give rise to several concepts in the theory of knots, surfaces and 3-manifolds. I will deal with some of them and show how they relate to each other.