Oxford

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).

The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.

In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d N = 4 supersymmetric gauge theories and explain how it leads to an exact calculation of quantum corrections to the Coulomb branch and a finite version of the AGT correspondence. 

Calabi-Yau spaces provide well-understood examples of supersymmetric vacua in supergravity. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood.
In this talk, I will review work that defines the analogue of Calabi-Yau geometry for generic D=4, N=2 supergravity backgrounds. The geometry is characterised by a pair of structures in generalised geometry, where supersymmetry is equivalent to integrability of the structures. I will also discuss the extension AdS backgrounds, where deformations of these geometric structures correspond to exactly marginal deformations of the dual field theories.

M5-branes on 4-manifolds M_4 realized as co-associatives in G_2 give rise to 2d (0,2) superconformal theories. In this talk I will propose a duality between these 2d (0,2) theories and 4d topological theories, which are sigma-models from M_4 into the Nahm moduli space. 

I will review the recently exposed connection between N=2 superconformal field theories in four dimensions and vertex operator algebras (VOAs). I will outline some general features of the VOAs that arise in this manner and describe the manner in which they reflect four-dimensional operations such as gauging and Higgsing. Time permitting, I will also touch on the modular properties of characters of these VOAs.

After mentioning, by way of motivation (mine at least), some diophantine questions concerning
sets definable in the restricted analytic, exponential field $\R_{an, exp}$, I discuss the
problem of extending a given $\R_{an, exp}$-definable function $f:(a, \infty) \to \R$ to
a holomorphic function $\hat f : \{z \in \C : Re(z) > b \} \to \C$ (for some $b > a$).
In particular, I give a necessary and sufficient condition on $f$ for such an $\hat f$ to exist and be
$\R_{an, exp}$-definable.
 

The large sieve is a powerful analytic tool in number theory, with many beautiful and diverse applications. In its most general form it resembles an approximate Bessel's inequality, and this clear modern theory rests on the combined effort of countless mathematicians in the mid-twentieth century -- Linnik, Roth, Selberg, Montgomery, Vaughan, and Bombieri, to name a few. However, it is hardly obvious to the beginner why this rather abstract inequality should be called 'large', or 'sieve'. In this introductory talk, aimed particularly at new graduate students, we discuss the rudimentary theory of the large sieve, some particular applications to sieving problems, and (at least one) proof. 

I will begin the talk by reviewing the definition of commutative K-theory, a generalized cohomology theory introduced by Adem and Gomez. It is a refinement of topological K-theory, where the transition functions of a vector bundle satisfy a commutativity condition. The theory is represented by an infinite loop space which is called a “classifying space for commutativity”.  I will describe the homotopy type of this infinite loop space. Then I will discuss the graded ring structure on its homotopy groups, which corresponds to the tensor product of vector bundles.