# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T. Consider the homology H_*(M) over some ring R.

Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a "projective linear” version M^{pl} of the moduli stack M.

For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra.

The construction appears to be new, but is connected with a lot of work in Geometric Representation Theory, to do with Ringel-Hall-type algebras and their representations, such as the results of Grojnowski-Nakajima on Hilbert schemes. The vertex algebra construction is enormously general, and applies in huge classes of examples. There is a differential-geometric version too.

The question I am hoping someone in the audience will answer is this: what is the physical interpretation of these vertex algebras?

It is in some sense an "even Calabi-Yau” construction: when applied to coh(X) or D^bcoh(X), it is most natural for X a Calabi-Yau 2-fold or Calabi-Yau 4-fold, and is essentially trivial for X a Calabi-Yau 3-fold. I discovered it when I was investigating wall-crossing for Donaldson-Thomas type invariants for Calabi-Yau 4-folds. So perhaps one should look for an explanation in the physics of Calabi-Yau 2-folds or 4-folds, with M the moduli space of boundary conditions for the associated SCFT.

Part of the series 'What do historians of mathematics do?'

We consider a vector-valued function $f: \mathbb{R}_+ \to X$ which is locally of bounded variation and give a decay rate for $|A(t)|$ for increasing $t$ under certain conditions on the Laplace-Stieltjes transform $\widehat{dA}$ of $A$. For this, we use a Tauberian condition inspired by the work of Ingham and Karamata and a contour integration method invented by Newman. Our result is a generalisation of already known Tauberian theorems for bounded functions and is applicable to Dirichlet series. We will say something about the connection between the obtained decay rates and number theory.

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.

In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.