In 1953 Yu. Prohorov published a paper on weak convergence of
probability measures on metric spaces, bringing a new, extended context to the Invariance Principle proved by Donsker two years earlier.
Prohorov���s formalism, publicised in books by K.R. Parthasarathy and
P. Billingsley, established the equivalence of the notion of convergence
in law of stochastic processes and the weak convergence of their distributions. This point of view is completely justified in metric spaces,
especially in Polish spaces.

It is, however, much less satisfactory in non-metric spaces, as was
shown by examples due to X. Fernique, given long time ago.
We show that in a large class of submetric spaces there exists a
stronger mode of convergence, coinciding with the weak convergence
on metric spaces, and much more suitable for needs of contemporary
theory of stochastic partial differential equations.

A submetric space is a topological space (X,tau) admitting a continuous metrics d that in turn determines a metric topology $\tau_d \subset \tau$ (where this inclusion is in general strict). As a standard (and the simplest)
example may serve a separable Hilbert space equipped with the weak
topology.

In these series of lectures we will explore initial value problem in general relativity and how it can be solved in a computer in practical situations. We will first cover the necessary mathematical foundations, including the concepts of well-posedness and strong hyperbolicity, and then explore the current formulations of Einstein��������s theory of gravity that are implemented in modern numerical codes, namely generalised harmonic coordinates and the BSSN formulation. We shall see how the latter can be implemented in a toy code so as to get some hands on experience. Time permitting, we will also explore the initial boundary value problem in asymptotically anti-de Sitter spaces and how it can be solved in practice using the characteristic formulation of the Einstein equations in applications of holography.

I will discuss a recent progress on two classical problems. The first one comes mostly from applied mathematics and numerical analysis: find tight universal and preferably simple enclosures for zeros of Bessel functions, of their derivatives, and possibly of other special functions. The second one comes primarily from number theory: find bounds for the number of lattice points under the graph of a given function (with some restrictions on the class of functions). As an application of these results, I���ll show the validity of inequalities �� la P��lya for the magnetic Aharonov--Bohm Laplacian in the disk, discuss possible generalisations, and open problems. The talk covers some joint works, mostly in progress, with N. Filonov, I. Polterovich, and D. A. Sher.

We will consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy's variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this talk we will discuss the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA.

In particular we analyse the stability of Cardy's ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. Based on arXiv:2306.13719.

King's College London Mathematics School is for students aged 16-18 with an enthusiasm and aptitude for mathematics, and aims to widen participation in high-quality degrees and careers in the mathematical sciences.

The school opened in 2014 and for nearly 10 years has been evolving a pedagogy and curriculum tuned to generating confident, skilled and articulate mathematical thinkers.

In this talk I will explore those aspects of pedagogy and culture at KCLMS that universities might use to improve their teaching and learning.

Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if Y is a homology 3-sphere other than S^3, then its fundamental group admits a homomorphism to SL(2,C) with non-abelian image. In this talk, I���ll explain how to generalize this to any Y whose first homology is 2-torsion or 3-torsion, other than the connect sum of n copies of the three-dimensional real projective space for any n or lens spaces of order 3. This is joint work with Sudipta Ghosh and Raphael Zentner.

Random permutations show up in a variety of areas in mathematics and its applications. In connection with physical applications (e.g., the lambda transition for superfluid helium), there is an interest in random spatial permutations -- that is, laws on permutations that have a 'geometric bias'. There are compelling heuristic arguments that this spatial bias has little effect on the distribution of the largest cycles of a random spatial permutation, provided that large cycles actually exist. I'll discuss a particular model of random spatial permutations (directed permutations on asymmetric tori) where these heuristics can be made precise, and large cycles can be shown to follow the expected (Poisson-Dirichlet) law.

In these series of lectures we will explore initial value problem in general relativity and how it can be solved in a computer in practical situations. We will first cover the necessary mathematical foundations, including the concepts of well-posedness and strong hyperbolicity, and then explore the current formulations of Einstein��������s theory of gravity that are implemented in modern numerical codes, namely generalised harmonic coordinates and the BSSN formulation. We shall see how the latter can be implemented in a toy code so as to get some hands on experience. Time permitting, we will also explore the initial boundary value problem in asymptotically anti-de Sitter spaces and how it can be solved in practice using the characteristic formulation of the Einstein equations in applications of holography.