Talk

The quest for solid foundations for $\infty$-categories started several decades ago, but from the mid-2000s there have been spectacular developments which excitingly accelerated the pace of progress. This workshop will focus on these objects, both in their foundations and in the applications to different fields of mathematics. There will be minicourses on $\infty$-categories, and also on $\infty$-operads, on algebraic K-theory and on parametrized homotopy theory, all three of which have benefited substantially from the $\infty$-categorical viewpoint.

I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism.

Using the Dolbeault picture of the moduli space of Higgs bundles, we will construct a function from the moduli space to an affine space.This function will be the Hamiltonian of an integrable system called the Hitchin system. This by definition gives a foliation of the moduli space generically by open subsets of abelian varieties. Time permitting, we will see some applications and how this leads to Higgs bundles offering new, and some times unexpected, geometric insight.

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

Talk 1: Title: Singular Hodge theory of matroids

If you take a collection of planes in R^3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.

On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it.

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

Given a positive definite integral binary quadratic form, it is a classical problem in number theory to count the integers that are represented by this form. A modern treatment was given in 2006 by Valentin Blomer and Andrew Granville.