University of Edinburgh

In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures.  

We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. 

In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  

In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. 

We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow. 

This is based on joint works with  J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).

Numerically solving PDEs typically requires compressing infinite information into a finite system of algebraic equations. Pragmatically, we usually follow a recipe: “Assume solutions of form X; substitute into PDE Y; discard terms by rule Z.” In contrast, Lanczos’s pioneering “tau method” prescribes modifying the PDE to form an exact finite system. Crucially, any recipe-based method can be viewed as adding a small equation correction, enabling us to compare multiple schemes independently of the solver. 

This talk also addresses a paradox: PDEs often admit infinitely many solutions, but finite systems produce only a finite set. When we include a “small” correction, the missing solutions are effectively hidden. I will discuss how tau methods frame this perspective and outline proposals for systematically studying and optimising various residuals.

The objects of arithmetic geometry are not manifolds. Some concepts from differential geometry admit analogues in arithmetic, but they are not straightforward. Nevertheless, there is a growing sense that the right way to understand certain Langlands phenomena is to study quantum field theories on these objects. What hope is there of making this thought precise? I will propose the beginnings of a mathematical framework via a general theory of factorization algebras. A new feature is a subtle piece of additional structure on our objects – what I call an _isolability structure_ – that is ordinarily left implicit.

The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g. It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^* . In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory. I will construct an induced module from an element of W^* and show that its annihilator is a primitive ideal. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context. In particular, we discover new bosonizations of q-W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

Deep Gaussian processes have proved remarkably successful as a tool for various statistical inference tasks. This success relates in part to the flexibility of these processes and their ability to capture complex, non-stationary behaviours. 

In this talk, we will introduce the general framework of deep Gaussian processes, in which many examples can be constructed, and demonstrate their superiority in inverse problems including computational imaging and regression.

 We will discuss recent algorithmic developments for efficient sampling, as well as recent theoretical results which give crucial insight into the behaviour of the methodology.

 

I will discuss certain dynamics of interacting particles in interlacing arrays with inhomogeneous, in space and time, jump probabilities and their relations to conditioned random walks and random tilings of the Aztec diamond.

In this talk, we will present a loop expansion for lattice gauge theories and its application to prove ultraviolet stability in the Abelian Higgs model. We will first describe this loop expansion and how it relates to earlier works of Brydges-Frohlich-Seiler. We will then show how the expansion leads to a quantitative diamagnetic inequality, which in turn implies moment estimates, uniform in the lattice spacing, on the Holder-Besov norm of the gauge field marginal of the Abelian Higgs lattice model. Based on Gauge field marginal of an Abelian Higgs model, which is joint work with Ajay Chandra.

The study of cohomology of infinite-dimensional Lie algebras was started by Gel'fand and Fuchs in the late 1960s. Since then, significant progress has been made, mainly focusing on the Witt algebra (the Lie algebra of vector fields on the punctured affine line) and some of its subalgebras. In this talk, I will explain the basics of Lie algebra cohomology and sketch the computation of the first cohomology group of certain subalgebras of the Witt algebra known as submodule-subalgebras. Interestingly, these cohomology groups are, in some sense, controlled by the cohomology of the Witt algebra. This can be explained by the fact that the Witt algebra can be abstractly reconstructed from any of its submodule-subalgebras, which can be described as a universal property satisfied by the Witt algebra.