Mon, 02 Nov 2020
12:45
Virtual

### Instantons, symmetries and anomalies in five dimensions

Pietro Benetti Genolini
(University of Cambridge)
Abstract

All five-dimensional non-abelian gauge theories have a U(1)U(1)I​U(1) global symmetry associated with instantonic particles. I will describe a mixed ’t Hooft anomaly between this and other global symmetries of  the theory, namely the one-form center symmetry or ordinary flavor symmetry for theories with fundamental matter. I will explore some general dynamical properties of the candidate phases implied by the anomaly, and apply our results to supersymmetric gauge theories in five dimensions, analysing the symmetry enhancement patterns occurring at their conjectured RG fixed points.

Thu, 04 Jun 2020

16:00 - 16:45
Virtual

### Expanders and generalisations

Ana Khurkho
(University of Cambridge)
Further Information

Part of the UK virtual operator algebras seminar: https://sites.google.com/view/uk-operator-algebras-seminar/home

Abstract

After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

Fri, 12 Jun 2020

14:00 - 15:00
Virtual

### Live-modelling the temporal regulation of mesoderm specification

Dr Berta Verd
(University of Cambridge)
Abstract

Pattern formation emerges during development from the interplay between gene regulatory networks (GRNs) acting at the single cell level and cell movements driving tissue level morphogenetic changes. As a result, the timing of cell specification and the dynamics of morphogenesis must be tightly cross-regulated. In the developing zebrafish, mesoderm progenitors will spend varying amounts of time (from 5 to 10hrs) in the tailbud before entering the pre-somitic mesoderm (PSM) and initiating a stereotypical transcriptional trajectory towards a mesodermal fate. In contrast, when dissociated and placed in vitro, these progenitors differentiate synchronously in around 5 hours. We have used a data-driven mathematical modelling approach to reverse-engineer a GRN that is able to tune the timing of mesodermal differentiation as progenitors leave the tailbud’s signalling environment, which also explains our in vitro observations. This GRN recapitulates pattern formation at the tissue level when modelled on cell tracks obtained from live-imaging a developing PSM. Our “live-modelling” framework also allows us to simulate how perturbations to the GRN affect the emergence of pattern in zebrafish mutants. We are now extending this analysis to cichlid fishes in order to explore the regulation of developmental time in evolution.

Thu, 13 Feb 2020

13:00 - 14:00
N3.12

### Ethics in Mathematics - where to begin?

Maurice Chiodo
(University of Cambridge)
Abstract

In recent years it has become abundantly clear that mathematics can do "things" in society; indeed, many more things than in the past. Deep mathematical work now underpins some of the most important aspects of the way society functions. And, as mathematically-trained people, we are constantly promoting the positive impact of mathematics. But if such work is capable of good, then is it not also capable of harm? So how do we begin to identify such potential harm, let alone address it and try and avoid, or at least reduce, it? In this session we will discuss how mathematics is a powerful double-edged sword, and why it must be wielded responsibly.

Thu, 23 Jan 2020
12:00
L4

### Vanishing viscosity limit of the compressible Navier-Stokes equations with general pressure law

Simon Schulz
(University of Cambridge)
Abstract

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

Thu, 12 Mar 2020

14:00 - 15:00
L4

### The Statistical Finite Element Method

Mark Girolami
(University of Cambridge)
Abstract

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

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Fri, 13 Mar 2020

14:00 - 15:00
L6

### Lava flows: theory, laboratory experiments and field data

Herbert Huppert
(University of Cambridge)
Abstract

World wide, unconstrained lava flows kill people almost each year and cause extensive damage, costing millions of pounds. Defending against lava flows is possible by using topographic variations sensibly, placing buildings considerately, constructing defending walls of appropriate size and the like. Hinton, Hogg and Huppert have recently published three rather mathematical papers outlining how viscous flows down slopes interact with a variety of geometrical shapes; evaluating, in particular, the conditions under which “dry zones” form – safe places for people and belongings – and the size of a protective wall required to defend a given size building.

Following a desktop experimental demonstration, we will discuss these analyses and their consequences.

Tue, 05 Nov 2019

14:00 - 15:00
L6

### Combinatorial discrepancy and a problem of J.E. Littlewood

Julian Sahasrabudhe
(University of Cambridge)
Further Information

Given a collection of subsets of a set X, the basic problem in combinatorial discrepancy theory is to find an assignment of 1,-1 to the elements of X so that the sums over each of the given sets is as small as possible. I will discuss how the sort of combinatorial reasoning used to think about problems in combinatorial discrepancy can be used to solve an old conjecture of J.E. Littlewood on the existence of flat Littlewood polynomials''.

This talk is based on joint work with Paul Balister, Bela Bollobas, Rob Morris and Marius Tiba.

Thu, 28 Feb 2019

16:00 - 17:00
L6

### Arithmetic statistics via graded Lie algebras

Beth Romano
(University of Cambridge)
Abstract

I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

Thu, 21 Feb 2019

16:00 - 17:00
L6

### GCD sums and sum-product estimates

Aled Walker
(University of Cambridge)
Abstract

When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares.

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