A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here I will discuss a proposal for a `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. In particular I will discuss how in maximally chaotic systems there is a suppression of exponential growth in commutator squares of generic few-body operators. This suppression appears to indicate that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems.

Given two probability distributions in R^d, a transport map is a function which maps samples from one distribution into samples from the other. For absolutely continuous measures, Brenier proved a remarkable theorem identifying a unique canonical transport map, which is "monotone" in a suitable sense. We study the question of whether this map can be efficiently estimated from samples. The minimax rates for this problem were recently established by Hutter and Rigollet (2021), but the estimator they propose is computationally infeasible in dimensions greater than three. We propose two new estimators---one minimax optimal, one not---which are significantly more practical to compute and implement. The analysis of these estimators is based on new stability results for the optimal transport problem and its regularized variants. Based on joint work with Manole, Balakrishnan, and Wasserman and with Pooladian.

The alchemists wanted to create gold, Hilbert wanted an algorithm to solve Diophantine equations, researchers want to make deep learning robust in AI, MATLAB wants (but fails) to detect when it provides wrong solutions to linear programs etc. Why does one not succeed in so many of these fundamental cases? The reason is typically methodological barriers. The history of  science is full of methodological barriers — reasons for why we never succeed in reaching certain goals. In many cases, this is due to the foundations of mathematics. We will present a new program on methodological barriers and foundations of mathematics,  where — in this talk — we will focus on two basic problems: (1) The instability problem in deep learning: Why do researchers fail to produce stable neural networks in basic classification and computer vision problems that can easily be handled by humans — when one can prove that there exist stable and accurate neural networks? Moreover, AI algorithms can typically not detect when they are wrong, which becomes a serious issue when striving to create trustworthy AI. The problem is more general, as for example MATLAB's linprog routine is incapable of certifying correct solutions of basic linear programs. Thus, we’ll address the following question: (2) Why are algorithms (in AI and computations in general) incapable of determining when they are wrong? These questions are deeply connected to the extended Smale’s 9th and 18th problems on the list of mathematical problems for the 21st century. 

Volumetric X-ray tomography is used in many areas, including applications in medical imaging, many fields of scientific investigation as well as several industrial settings. Yet complex X-ray physics and the significant size of individual x-ray tomography data-sets poses a range of data-science challenges from the development of efficient computational methods, the modelling of complex non-linear relationships, the effective analysis of large volumetric images as well as the inversion of several ill conditioned inverse problems, all of which prevent the application of these techniques in many advanced imaging settings of interest. This talk will highlight several applications were specific data-science issues arise and showcase a range of approaches developed recently at the University of Southampton to overcome many of these obstacles.

There are various aspects of the AdS/CFT correspondence that are rather mysterious. For example, how does the gravitational theory know about a discrete boundary spectrum or how does it know moments of the partition function factorize, given the existence of connected (wormhole) geometries? In this talk I will discuss some recent efforts with Andreas Blommaert and Luca Iliesiu on these two puzzles in two dimensional dilaton gravities. These gravity theories are simple enough that we can understand and propose a resolution to the discreteness and factorization puzzles. I will show that a tiny but universal bilocal spacetime interaction in the bulk is enough to ensure factorization, whereas modifying the dilaton potential with tiny corrections gives a discrete boundary spectrum. We will discuss the meaning of these corrections and how they could be related to resolutions of the same puzzles in higher dimensions. 

In this talk we discuss the set of generalized symmetries associated with the free graviton theory in four dimensions. These are generated by ring-like operators. As for the Maxwell field, we find a set of “electric” and a dual set of “magnetic” topological operators and compute their algebra. The associated electric and magnetic fields satisfy a set of constraints equivalent to the ones of a stress tensor of a 3d CFT. This implies that the generalized symmetry is charged under space-time symmetries, and it provides a bridge between linearized gravity and the tensor gauge theories that have been introduced recently in the context of fractonic systems in condensed matter physics.

Preparing for Prelims and Part A exams with Dr Vicky Neale

Description: This session will offer guidance for Prelims and Part A students preparing for closed-book, in-person exams this summer, with tips on revision and information about practical arrangements. If you have questions, please send them in advance (by 28 February) via https://vevox.app/#/m/170975861 and we'll try to address as many as possible during the session.

A separate session in Week 6 will be aimed at students doing Part B, Part C and MSc exams.

In many contexts a correspondence has been found between the classical compact groups and certain boundary conditions -- $U(n)$ corresponding to periodic, $USp(2n)$ corresponding to Dirichlet, $SO(2n)$ corresponding to Neumann and $SO(2n+1)$ corresponding to Zaremba. In this talk, I will try to elucidate this correspondence in Lie theoretic terms and in the process relate random matrix theory to Yang-Mills theory, free fermions and modular forms.