Events

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their
structural properties. The course is aimed to be accessible by graduate students and young researchers.

Serre-Tate proved in the 60's that an ordinary abelian variety E over a finite field k can be canonically lifted to an ordinary abelian variety over the Witt vectors of k. Expressed in terms of the moduli stack M_ord of ordinary abelian schemes, this is precisely a universal lifting property with respect to the Witt vectors (of finite fields).

The aim of this talk is to give an overview of various subrings of the Chow ring of hyperKaehler varieties, which like the Beauville-Voisin ring conjecturally inject in cohomology via the cycle class map. I will give examples of hyperKaehler varieties for which such injectivity statements are known to be true and I will explain how these various subrings relate to each other (conjecturally and in examples). I will also concomitantly draw the parallel picture for abelian varieties. This is based on works with Lie Fu, Robert Laterveer and Mingmin Shen.

Toda systems are important examples of integrable systems, and we consider the version using the Laplacian on the plane. They are generalizations of the Liouville equation to simple Lie algebras. As such, they are related to 2-dimensional conformal geometry and harmonic maps. 

In this talk, we will discuss the classification of solutions to such Toda systems and the quantization result on their total masses. Such total masses have applications to the local blowup analysis of the  mean field equations.