**Send email to jeremy.mann@kcl.ac.uk for link to online seminar.** In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For an initial data set satisfying the vacuum constraint equations, there exists a solution to the vacuum Einstein equations and it is geometrically unique in the domain of dependence of the initial surface. On contrast, the initial boundary value problem (IBVP) has been much less understood. To solve for an vacuum metric in a region with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss obstacles in establishing a well-defined IBVP for vacuum Einstein equations and the geometric uniqueness problem. Then I will talk about an existence and geometric uniqueness result in a joint work with Michael Anderson.

The purpose of this talk is to introduce a practical method for assessing mathematical proof online. We examine the effect of faded worked examples and reading comprehension questions to student's understanding of proof. By breaking down a given proof, we will demonstrate a checklist that can be used to generate comprehension questions which can be assessed automatically online. We then provide some preliminary results of deploying such questions.
In addition, we shall examine several examples of proof comprehension questions that have been created by following this checklist. In particular, I will discuss some ongoing work related to the creation of proof comprehension tasks used to assessment of proof in a higher-level undergraduate pure mathematics courses.

In recent years, significant progress has been made in the field of birational geometry for foliations. Notably, the Minimal Model Program (MMP) has been shown to work for foliations on threefolds. In this talk, I will demonstrate that the MMP is applicable to toric foliations as well. Specifically, I will discuss how non-dicritical singularities (and foliated dlt singularities if time permits) are preserved under the MMP. This is a joint work with Chih-Wei Chang.

Random intersection graph is a simple random graph model that incorporates community structures. To build such a graph, imagine there are n individuals and m potential communities, where each individual joins a community independently with probability p. Two individuals are adjacent to each other in the random intersection graph G(n, m, p) if and only if they belong to a common community. It is known that if m>>n^3, then the model is asymptotically equivalent to an Erdos-Renyi graph G(n, q) with a suitable q. I���ll discuss other ways to compare the two models, particularly regarding the scaling limit of the graphs inside the critical window and number of triangles found in the large components.

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.

Model-based clustering represents one of the fundamental procedures in a statistician's toolbox. Within the model-based clustering framework, we consider the case where the kernel distribution of nonparametric mixture models is available only up to an intractable normalizing constant, in which most of the commonly used Markov chain Monte Carlo methods fail to provide posterior inference. To overcome this problem, we propose an approximate Bayesian computational strategy, whereby we approximate the posterior to avoid the intractability of the kernel. By exploiting the structure of the nonparametric prior, our proposal combines the use of predictive distributions as a proposal with transport maps to obtain an efficient and flexible sampling strategy. Further, we illustrate how the specification of our proposal can be relaxed by introducing an adaptive scheme on the degree of approximation of the posterior distribution. Empirical evidence from simulation studies shows that our proposal outperforms its main competitors in terms of computational times while preserving comparable accuracy of the estimates.

A ��������t Hooft loop is a typical disorder operator defined in any gauge theory that can be studied by a combination of holography, localization and integrability. After reviewing the quantum mechanics magnetic monopoles, I will describe how integrability and Bethe ansatz can help to study ��������t Hooft loops in the N=4 super-Yang-Mills theory.

Gaussian Processes and the Kullback-Leibler divergence have been deeply studied in Statistics and Machine Learning. This paper marries these two concepts and introduce the local Kullback-Leibler divergence to learn about intervals where two Gaussian Processes differ the most. We address subtleties entailed in the estimation of local divergences and the corresponding interval of local maximum divergence as well. The estimation performance and the numerical efficiency of the proposed method are showcased via a Monte Carlo simulation study. In a medical research context, we assess the potential of the devised tools in the analysis of electrocardiogram signals.

I will discuss sharp estimates for the norm of the operator ���identity minus conditional expectation���. They allow one to find the optimal constant in the bounded compact approximation property of Lp([0, 1]), 1 < p < infty. I will also discuss related open problems. The talk is based on a joint paper with T. Sharia.