# UMI workshop 2019

## Key Information

Dates: 30 and 31 May 2019

Venue: Room 145, 180 Queen’s Gate (Huxley Building), Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2RH

Organisers: Professor Richard Craster and Professor Pierre Degond

For general enquires: Anuj Sood

## Perspectives in cross-channel mathematical collaborations within the UMI Abraham de Moivre

### May 30-31 2019, Imperial College London

The French CNRS celebrates its 80th Anniversary this year, to celebrate this we are holding a workshop to showcase exciting mathematics. The Abraham de Moivre UMI is a joint CNRS and Imperial College initiative to promote the exchange of mathematical ideas and of staff between the UK and France. This workshop brings together the scientific advisory board, visitors to the UMI and academic staff at Imperial College for a “festival of mathematics”, a group of leading experts in their respective fields will give accessible talks about their research areas. We hope you will join us!

Poster for the workshop (PDF)‌
Poster for the public lecture (PDF)

#### Schedule

Morning of Thursday 30 May

10:00-10:40: Welcome and coffee

10:40-11:20: Pierre Pansu (Université Paris Sud) - An invitation to geometric group theory

11:20-12:00: Ulrike Tillmann (University of Oxford) - New directions in Algebraic Topology

12:00-13:45: Lunch

Afternoon of Thursday 30 May

13:45-14:25: Toby Gee (Imperial College London) - Potential modularity of genus 2 curves

14:25-14:40: Jean-Stéphane Dhersin (CNRS) - Opportunities within the UMI Abraham de Moivre

14:40-15:00: Coffee break

15:00-16:30: Meeting of the Scientific Advisory Board (Room 747, Huxley Building)

Evening of Thursday 30 May (please note that this session will take place in the Clore Lecture Theatre, Huxley Building)

18:00-19:00: Public lecture by Kevin Buzzard (Imperial College London) - The future of mathematics?

Morning of Friday 31 May

9:30-10:10: Martin Hairer (Imperial College London) - Universality classes for 1+1 dimensional systems

10:10-10:50: Anne Gégout-Petit (Université de Lorraine) - Inference for Piecewise Deterministic Markov Processes

10:50-11:20: Coffee break

11:20-12:00: Sébastien Guenneau (CNRS, Institut Fresnel, Marseille) - Metamaterials for the control of wave and diffusion phenomena: Models and experiments

12:00-14:00: Lunch

Afternoon of Friday 31 May

14:00-14:40: Pascal Noble (Institut National des Sciences Appliquées, Toulouse) - 2D versus 1D shallow water equations

14:40-15:20: Tom Coates (Imperial College London) - Towards a Periodic Table of Shapes

15:20-15:40: Meeting wrap-up

#### Book of Abstracts

Public lecture : Kevin Buzzard (Imperial College London)

Title : The future of mathematics?

Abstract : Computers have changed the way that humans do mathematics: they enable us to do experiments in an extremely efficient way. If a mathematician comes up with a new conjecture about numbers, they could test it in thousands or even millions of special cases without even leaving their office, before devoting any time trying to prove it in general. But there are infinitely many numbers, and a computer cannot check infinitely many things.

In the future, computers will change the way that humans do mathematics in a second way. They will help humans to look for proofs. I believe that advances in AI and machine learning, combined with growing databases of mathematical theorems, make this the inevitable future of mathematics. Computers can beat us at chess already -- when will they start to beat us at proving mathematical theorems? Will the jobs of research mathematicians be under threat? Should this change the way we teach mathematics?

I will give an introductory survey of the differences between the ways that humans and computers do mathematics, and how humans and computers are learning to work together. No technical background will be assumed and the talk will be suitable for a general audience.

Tom Coates (Imperial College London)

Title: Towards a Periodic Table of Shapes

Abstract: Fano manifolds are basic building blocks in algebraic geometry -- they are, in a precise sense, atomic pieces of mathematical shapes.  I will describe a project, joint work with Alessio Corti, Vasily Golyshev, Alexander Kasprzyk and many others, to find and classify Fano manifolds, using a circle of ideas from physics called Mirror Symmetry.

Toby Gee (Imperial College London)

Title : Potential modularity of genus 2 curves

Abstract : I will discuss, in very down to earth terms, some recent joint work with George Boxer, Frank Calegari, and Vincent Pilloni on the Hasse-Weil conjecture for curves of genus 2. In particular I will explain this work in terms of a completely elementary question about counting solutions to a polynomial equation. I will also explain a connection with the Riemann zeta function. Nothing beyond first year undergraduate mathematics will be assumed.

Anne Gégout-Petit (Université de Lorraine)

Title : Inference for Piecewise Deterministic Markov Processes

Abstract : Piecewise-deterministic Markov processes have been introduced in the literature by Davis (1993) as a general class of non-diffusion stochastic models. They are a family of Markov processes involving deterministic motion punctuated by random jumps, which occur either when the flow hits the boundary of the state space or in a Poisson-like fashion with nonhomogeneous rate. The path depends on three local characteristics namely the flow $\Phi$, the jump rate $\lambda$, which determines the interarrival times, and the transition kernel Q, which specifies the postjump location. Over the past decade, statisticians have been interested in estimating one or more of these three characteristics based on the observation of several trajectories or a trajectory observed over a long period of time. We will present some results around the inference on the PDMP, including our work on estimating the distribution of the process jump rate distribution which is a function of two variables: a spatial mark and time. The framework assumes that jumps are accurately observed. We will also discuss the detection of regimes in piecewise deterministic polynomial models.

Sébastien Guenneau (CNRS, Institut Fresnel, Marseille)

Title: Metamaterials for the control of wave and diffusion phenomena: Models and experiments

Abstract: Building upon analogies with flat lenses via negative refraction and invisibility cloaks for light, physicists and applied mathematicians have designed novel types of structured media, coined metamaterials, that can control miscellaneous types of waves in unprecedented ways. This talk will cover some of the interesting physics that occurs and then discuss the mathematical questions  that this poses.

Martin Hairer (Imperial College London)

Title: Universality classes for 1+1 dimensional systems

Abstract: We will discuss several toy models for interface fluctuations and their large-scale behaviour, with an emphasis on trying to understand the "big picture". On the way, we find a renormalisation fixed point with infinitely many unstable directions.

Pascal Noble (Institut National des Sciences Appliquées, Toulouse)

Title: 2D versus 1D shallow water equations

Abstract: In this talk, we will derive consistent 1D section averaged shallow water equations from 2D shallow water equations. This is done by carrying out an asymptotic expansion of the fluid and pressure fields in the quasi 1D limit. It is proved in particular that the velocity field is not constant in the crosstream direction which is a common assumption in the literature. We validate our model through numerical computations of backwater curves.

Pierre Pansu (Université Paris Sud)

Title: An invitation to geometric group theory

Abstract: Geometric group theory emerged as a separate branch of topology in the 1980s. It has close connections with dynamical systems, operator algebras, probability and certain aspects of theoretical computer science. I will sketch a portrait of this field which is well represented both in UK and in France.

Ulrike Tillmann (University of Oxford)

Title : New directions in Algebraic Topology

Abstract  Looking back on a century of history, the field has undergone many changes. Often thought of from the outside as a useful tool box its methods have been exported with great success to nearly all mathematical subject areas including one of its newest additions, mathematical data science. In the meantime, through its dialogue with other mathematical areas, algebraic topology it constantly adjusting and reinventing itself while not forgetting its own old challenges. We will give a bird's eye overview.