MINICURSOS



Noé Bárcenas (Centro de Ciencias Matemáticas, UNAM Morelia)
Título: Index theory, K-theory and equivariant algebraic topology
Resumen: In this lecture series we will give a gentle introduction to topological K-theory and the ideas of index theory related to the Baum-Connes conjecture with an equivariant algebraic topological flavor.

Andrew Blumberg (University of Texas at Austin)
Título: Homotopical categories
Resumen: This course will be an introduction to the homotopy category, homotopy (co)limits, cofibration categories, derived functors, model categories, etc., leading up to a little bit about why people are excited about infinity categories and how to think about the kinds of problems they solve.

Jesús González (Centro de Investigación y Estudios Avanzados del IPN)
Título: Topología algebraica y computacional en la robótica
Resumen: En los últimos 15 años se han desarrollado técnicas topologico-algebraicas para estudiar uno de los problemas centrales en la robótica: la planeación motriz de sistemas autónomos. En este curso se revisará el modelo de Farber (complejidad topológica) para dicho problema, presentando sus propiedades generales, así como los avances recientes en el área y líneas de investigación a futuro. El curso tratará de ser autocontenido para estudiantes de licenciatura. En particular, se revisarán los conceptos básicos necesarios en topología algebraica.

Mike Hill (Universidad de California, Los Ángeles)
Título: Classical and equivariant homotopy
Resumen: Homotopy theory studies topological spaces up to a weaker equivalence relation: homotopy. Many of the algebraic invariants we consider are homotopy invariants, so in many ways, this is an accessible and intuitive simplification. Many spaces which arise in nature, however, come equipped with the action of some group of symmetries, and we can ask that our relations preserve this extra structure. This is "equivariant homotopy theory". In this series of talks, I'll develop the basics of equivariant homotopy theory, explaining how one works with these objects and describing some of the unexpected issues.

Antonio Rieser (CONACYT-Centro de Investigación en Matemáticas, A.C.)
Título: Homotopy and homology on point clouds and combinatorial objects
Resumen: A nearly ubiquitous assumption in modern data analysis is that a given high-dimensional data set concentrates around a lower dimensional space. Recently, a great deal of attention has been focused on how to use point samples from a metric measure space to estimate the topological and geometric invariants of this lower dimensional space, and on applying the resulting algorithms to real data sets. Most techniques for studying the topology of data, however, in particular persistent homology, proceed by considering families of topological spaces which were in some way thicker than the original set. In this course, we instead show how a non-trivial homotopy theory may be constructed directly on sets of points. We then show that the same construction also applies to a variety of combinatorial objects, and give several computations for homotopy groups on point clouds, graphs, and simplicial complexes. We will then explore homology theories in this setting.