Thu, 11 May 2023
16:00
L5

Parity of ranks of abelian surfaces

Celine Maistret
(University of Bristol)
Abstract
Let K be a number field and A/K an abelian surface. By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K.
Assuming finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces under suitable local constraints. Using a similar approach, we show that for two elliptic curves E_1 and E_2 over K with isomorphic 2-torsion, the parity conjecture is true for E_1 if and only if it is true for E_2.
In both cases, we prove analogous unconditional results for Selmer groups.
Tue, 31 Jan 2023

14:00 - 15:00
L4

Hypercontractivity on compact Lie groups, and some applications

David Ellis
(University of Bristol)
Abstract

We present two ways of obtaining hypercontractive inequalities for low-degree functions on compact Lie groups: one based on Ricci curvature bounds, the Bakry-Emery criterion and the representation theory of compact Lie groups, and another based on a (very different) probabilistic coupling approach. As applications we make progress on a question of Gowers concerning product-free subsets of the special unitary groups, and we also obtain 'mixing' inequalities for the special unitary groups, the special orthogonal groups, the spin groups and the compact symplectic groups. We expect that the latter inequalities will have applications in physics.

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

Tue, 18 Oct 2022

15:30 - 16:30
L6

Universal characteristics of deep neural network loss surfaces from random matrix theory

Nick Baskerville
(University of Bristol)
Abstract

Neural networks are the most practically successful class of models in modern machine learning, but there are considerable gaps in the current theoretical understanding of their properties and success. Several authors have applied models and tools from random matrix theory to shed light on a variety of aspects of neural network theory, however the genuine applicability and relevance of these results is in question. Most works rely on modelling assumptions to reduce large, complex matrices (such as the Hessians of neural networks) to something close to a well-understood canonical RMT ensemble to which all the sophisticated machinery of RMT can be applied to yield insights and results. There is experimental work, however, that appears to contradict these assumptions. In this talk, we will explore what can be derived about neural networks starting from RMT assumptions that are much more general than considered by prior work. Our main results start from justifiable assumptions on the local statistics of neural network Hessians and make predictions about their spectra than we can test experimentally on real-world neural networks. Overall, we will argue that familiar ideas from RMT universality are at work in the background, producing practical consequences for modern deep neural networks.

 

Mon, 16 May 2022

16:00 - 17:00
C1

TBA

Emilia Alvarez
(University of Bristol)
Mon, 07 Feb 2022

12:45 - 13:45
Virtual

On systems of maximal quantum chaos

Mike Blake
(University of Bristol)
Further Information

Note the unusual time and date

Abstract

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.

Tue, 08 Feb 2022

15:30 - 16:30
Virtual

Non-intersecting Brownian motion and compact Lie groups

Alex Little
(University of Bristol)
Abstract

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.

Tue, 15 Feb 2022

15:30 - 16:30
Virtual

A handful of moment computations of characteristic polynomials and their derivatives in the classical compact ensembles

Emilia Alvarez
(University of Bristol)
Abstract

I will present a collection of moment computations over the unitary, symplectic and special orthogonal matrix ensembles that I've done throughout my thesis. I will focus on the methods used, the motivation from number theory, the relationship to Painlev\'e equations, and directions for future work.

Mon, 11 Oct 2021

14:15 - 15:15
L4

Minimal surfaces, spectral geometry and homogenisation

Jean Lagacé
(University of Bristol)
Abstract

Free boundary minimal surfaces are a notoriously elusive object in geometric analysis. From 2011, Fraser and Schoen's research program found a relationship between free boundary minimal surfaces in unit balls and metrics which maximise the first nontrivial Steklov eigenvalue. In this talk, I will explain how we can adapt homogenisation theory, a branch of applied mathematics, to a geometric setting in order to obtain surfaces with first Steklov eigenvalue as large as possible, and how it leads to the existence of free boundary minimal surfaces which were previously thought not to exist.

Tue, 26 Jan 2021

15:30 - 16:30
Virtual

Secular coefficients and the holomorphic multiplicative chaos

Joseph Najnudel
(University of Bristol)
Abstract

We study the coefficients of the characteristic polynomial (also called secular coefficients) of random unitary matrices drawn from the Circular Beta Ensemble (i.e. the joint probability density of the eigenvalues is proportional to the product of the power beta of the mutual distances between the points). We study the behavior of the secular coefficients when the degree of the coefficient and the dimension of the matrix tend to infinity. The order of magnitude of this coefficient depends on the value of the parameter beta, in particular, for beta = 2, we show that the middle coefficient of the characteristic polynomial of the Circular Unitary Ensemble converges to zero in probability when the dimension goes to infinity, which solves an open problem of Diaconis and Gamburd. We also find a limiting distribution for some renormalized coefficients in the case where beta > 4. In order to prove our results, we introduce a holomorphic version of the Gaussian Multiplicative Chaos, and we also make a connection with random permutations following the Ewens measure.

Thu, 21 Jan 2021

12:00 - 13:30
Virtual

Node-based approximation of contagion dynamics on networks

Cameron Hall
(University of Bristol)
Abstract

Contagion models on networks can be used to describe the spread of information, rumours, opinions, and (more topically) diseases through a population. In the simplest contagion models, each node represents an individual that can be in one of a number of states (e.g. Susceptible, Infected, or Recovered), and the states of the nodes evolve according to specified rules. Even with simple Markovian models of transmission and recovery, it can be difficult to compute the dynamics of contagion on large networks: running simulations can be slow, and the system of master equations is typically too large to be tractable.

 One approach to approximating contagion dynamics is to assume that each node state is independent of the neighbouring node states; this leads to a system of ODEs for the node state probabilities (the “first-order approximation”) that always overestimates the speed of infection spread. This approach can be made more sophisticated by introducing pair approximations or higher-order moment closures, but this dramatically increases the size of the system and slows computations. In this talk, I will present some alternative node-based approximations for contagion dynamics. The first of these is exact on trees but will always underestimate the speed of infection spread on a network with loops. I will show how this can be combined with the classic first-order node-based approximation to obtain a node-based approximation that has similar accuracy to the pair approximation, but which is considerably faster to solve.

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