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.

Bayesian analyses combine information represented by different terms in a joint Bayesian model. When one or more of the terms is misspecified, it can be helpful to restrict the use of information from suspect model components to modify posterior inference. This is called ���cutting feedback���, and both the specification and computation of the posterior for such ���cut models��� is challenging. In this paper, we define cut posterior distributions as solutions to constrained optimization problems, and propose variational methods for their computation. These methods are faster than existing Markov chain Monte Carlo (MCMC) approaches by an order of magnitude. It is also shown that variational methods allow for the evaluation of computationally intensive conflict checks that can be used to decide whether or not feedback should be cut. Our methods are illustrated in examples, including an application where recent methodological advances that combine variational inference and MCMC within the variational optimization are used.

The talk will outline the main properties of high-dimensional random matrices whose entries do not have the 4th moment. We will also discuss heavy-tailed random matrices obtained from stock returns.

Undergraduate mathematics students see a lot of written proofs. But how much do they learn from them? Perhaps not as much as we would like, and perhaps we would like to improve their ability to engage with mathematics by reading. This talk will present a sequence of research studies addressing this issue. It will first describe studies on e-Proofs, which attempted to improve comprehension via proof-specific learning resources. It will then describe an eye-tracking study that provided real-time data on both student and expert mathematical reading. Finally, it will describe the effects of generic mathematical self-explanation training, investigated via both experimental and eye-tracking methods. Together, these studies provide insight into what is special about mathematical reading, and how students can be supported in reading more effectively.

Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman. We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will prove that non-archimedean periods recover the analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.

At the boundary of (discrete) probability and statistical physics, random walks in dynamic environments represent a way to model a particle advected by a fluid. Though some models are quite simple to define (usually, the environment is an evolving particle system, such as the exclusion process or a "cloud" of random walks), interest for this topic among probabilists has strongly increased in the past decade. These models feature complex technical challenges, as the time-dependency of the environment totally reshuffles the structure of space-time correlations, compared to the more classical setting of walks in static environment. In this talk, we will present this context and highlight how some key features such as recurrence/transience or the fluctuations of the walker are still the subject of many conjectures.

We study ensembles of 1/2-BPS bound states of fundamental strings and NS-fivebranes (NS5-F1) in the decoupling limit. We revisit a solution corresponding to an ensemble average of these bound states, and find that the appropriate duality frame for describing the near-source structure is the T-dual NS5-P frame.
When members of the ensemble spin with two fixed angular potentials about two orthogonal planes, our main result is that an ensemble average of them gives rise to a smooth horizonless solution characterized by an ellipsoidal structure. This contrasts with ring structures obtained when fixing the angular momenta instead of the angular potentials\DSEMIC we trace this difference of ensembles to large fluctuations of the angular momentum in the ensemble of fixed angular potential.