Oxford

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

Martin Bridson will give a "repeat" performance of his Abel Lecture which he delivered a few weeks ago in Oslo as part of the scientific programme in honour of Abel Prize laureate Mikhail Gromov.

Abstract:

Gromov has illuminated great swathes of mathematics with the bright light of geometry. By means of example, I hope to convey the sense of wonder that his work engenders and something of the profound influence he has had on the way my generation thinks about mathematics.

I shall focus particularly on Geometric Group Theory. Gromov's ideas turned the study of discrete groups on its head, infusing it with an array of revolutionary ideas and unveiling deep connections to many other branches of mathematics.

We generalize rings, Banach algebras and C*-algebras to ringoids, Banach algebroids and C*-algebroids. We construct algebraic and topological K-theory of these objects. As an application we can formulate Farrell-Jones Conjecture in algebraic K-theory, Bost- and Baum-Connes-Conjecture in topological K-theory

I shall discuss joint work with Angus Macintyre on the model theory of the ring of adeles of a number field

I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.

In this talk I will outline the two constructions of the Brauer group Br($X$) of a scheme $X$, namely via etale cohomology and Azumaya algebras and briefly describe how one may compute this group using the Hochschild-Serre spectral sequence. In the early '70s Manin observed that one can use the Brauer group of a projective variety $X/k$ to define an obstruction to the existence of rational points on $X$. I will discuss this arithmetic application and time permitting, outline an example for $X$ a K3 surface.