# Warwick-Imperial Autumn Meeting

#### 1st December 2018

Organisers: Rebecca Myers (Warwick), Rufus Lawrewnce(Imperial)

WIMP is a free one day conference primarily aimed at 3rd/4th year undergraduate, masters, and early PhD students from Warwick and Imperial though of course others are welcome. It is on Saturday 1st December in the Zeeman building, University of Warwick. The aim of the conference is to introduce different areas of mathematics that students are researching into, as well as featuring an academic's talk. We commence with a plenary talk and then divide into two streams of five talks, so there should be something of interest to everyone. All talks will be approximately 45 minutes in length with 15 minutes afterward for questions, discussion, and refreshment.

To register for the conference, please fill out the following form: WIMP2018-Registration-Form. This is helpful for booking and food organisation.

If you have any other questions please email r.myers.1@warwick.ac.uk

**Getting to campus:** For general advice on how to get to the campus see here https://www2.warwick.ac.uk/study/undergraduate/visits/gettinghere/

**Directions to WIMP:** For those new to the campus, the day starts inside the Warwick Mathematics Institute (map: https://tinyurl.com/yczddkj8). When you enter the building there is a staircase in front, slightly to the right. Go up the first flight of stairs and turn left. There is a common room where you can register.

**Schedule for the day: **(TBC-Some changes may occur)

**09:30:** Registration (Upstairs common room: Zeeman Building)

**10:00-11:00**:Plenary

[MS.04] Professor Dwight Barkley (Warwick) -Solving the 130 year old problem of turbulence

**11:00-12:00: **

[MS.04] Marco Barberis (Warwick) - Why Earth cannot be flat? (An introduction to surface geometry)

[MS.05] George Wynne (Currently: Imperial. Previous: Warwick) - Whirlwind explanation of probability theory and Gaussian processes and why you should care.

**12:00-13:00:**

[MS.04] Tudor Ciurca (Imperial)- Kummer Theory with applications to Fermat's Last Theorem

[MS.05] Tasos Stylianou (Warwick)- Packing measure of Gromov–Hausdorff space

**13:00-14:00**:

Lunch

**14:00-15:00**:

[MS.04] Lambert A'Campo (Imperial)- Dirichlet's Theorem on primes in arithmetic progressions

[MS.05] Emma Southall (Warwick)- Forseeing critical transitions in stochastic infectious disease models

**15:00-16:00**:

[MS.04] Harry Gouldbourne (Imperial)- The McKay Correspondence and McKay Graphs.

[MS.05] Rufus Lawrence (Imperial)- Pointless topology and how to fix measure theory

**16:00-16:30**:

Afternoon tea and coffee

**16:30-17:30**:

[MS.04] Harry Petyt (Currently: Bristol. Previous: Warwick)- Introduction to Tropical Geometry

[MS.05] Elliot Macneil (Imperial) -Formalising mathematics with automated theorem provers

**17:30**: (Common room)

**Abstracts:**

**Professor Dwight Barkley** (Warwick)- Solving the 130 year old problem of turbulence

More than a century ago Osborne Reynolds launched the study of turbulence as he sought to understand the conditions under which fluid flowing through a pipe would be smooth (laminar) or complex (turbulent). For years this problem has been the subject of study, controversy, and uncertainty. I will recall some of the history of linear stability analysis and chaos with a view to explaining why even the simplest case is both fascinating and difficult. I will then explain how exploiting results from different fields we have recently begun to solve this long-standing problem.

**Marco Barberis** (Warwick) -Why Earth cannot be flat? (An introduction to surface geometry)

In this talk I'd like to introduce the idea of geometry as a superstructure to topology. I will present geometric models of the plane, the sphere and the torus, and try to discuss why two spaces which are topologically identical can be extremely different from a geometric viewpoint. The talk will then focus on the concept of curvature, as a way to "measure" what the geometry of a space is, and introduce what the terms flat, spherical or hyperbolic mean to a geometer. In the end we will return to topology to investigate how this influence the possible geometry by stating Gauss-Bonnet Theorem.

For the entire duration of the talk we will mostly deal with spaces embedded in R^3, trying to keep thing as visual as possible, so nothing more that some basic calculus and the first definitions in topology should be required to follow the talk.

**George Wynne** (Imperial)- Whirlwind explanation of probability theory and Gaussian processes and why you should care.

My talk will be about Gaussian Processes, which are random functions with some nice properties. I will explain that at a high level the study of Gaussian Processes, indeed of many topics in probability theory, boils down to real analysis, functional analysis and measure theory arguments. In particular, I will talk about the use of Gaussian Processes in regression and optimisation and if I have time I will give more in depth details about the other parts of maths which they relate to. I will also aim to show that at graduate level studying statistics might not be what you think it is… This talk will assume basic real analysis and measure theory knowledge but knowledge of probability theory is not required!

**Tudor Ciurca** (Imperial)- Kummer Theory with applications to Fermat's Last Theorem

All quadratic extensions of Q are formed by adding a square root of a square-free integer. Kummer theory provides a generalization of this characterization. It parametrizes certain cyclic extensions through roots of elements in the base field. We will establish this using the technical but simple result “Hilbert’s theorem 90”. On the other hand, class field theory provides a parametrization of finite abelian extensions of a number field through data also internal to the base field. When studying certain extensions of cyclotomic fields, these two parametrizations interact in a wonderful way. This interaction will enable us to partially prove the reflection theorem. Kummer proved Fermat’s last theorem for all regular prime exponents. These are primes which satisfy a technical condition involving the ideal class groups of cyclotomic rings. As an application of our result, we get a much more computable condition for regular primes, called Kummer’s criterion.

**Tasos Stylianou** (Warwick)- Dimensions of a typical compact metric space

Many different notions of dimension were introduced in the 20th century to study the complexity of fractals. In this talk, we will present a brief introduction to the Box, Hausdorff and Packing dimension. In particular, we will calculate the packing dimension of a typical compact metric space belonging to the Gromov-Hausdorff space.

**Lambert A'Campo** (Imperial)- Dirichlet's Theorem on primes in arithmetic progressions

Dirichlet’s theorem states that for coprime positive integers a, N there are infinitely many primes of the form a + N k, k ∈ N. In fact we will even prove that the density of such primes is 1/φ(N), where φ(N) = |(Z/NZ) ×| is the euler totient function. The proof we present is one of the first examples of the use of analytic methods in number theory. The key insight will be to show that the theorem follows from the non-vanishing of certain holomorphic functions (Dirichlet L-functions) at 1. Actually proving that these functions don’t vanish is – maybe surprisingly – the hardest part of the proof.

**Emma Southall** (Warwick)- Forseeing critical transitions in stochastic infectious disease models

Bifurcations in a complex dynamical system can be predicted prior to reaching the threshold. This can be achieved by detecting statistical indicators as the system undergoes the phenomenon of critical slowing down. Indicators of critical slowing down theory can anticipate zero eigenvalue bifurcations and have previously been applied to many real-world systems from the global climate system to financial markets. This talk will present potential indicators of disease elimination for metapopulation models. A metapopulation is a network of interacting populations where the infection dynamics are influenced by events within populations and by spatial processes between connected populations.

**Harry Gouldbourne** (Imperial)- The McKay Correspondence and McKay Graphs.

The McKay correspondence is a statement in group representation theory, with interpretations in algebraic geometry. It states that “The McKay graphs of finite subgroups of SU(2) corresponding to the 2-dimensional characters are in bijection with the affine extended Dynkin diagrams”. I will start by looking at the finite subgroups of SU(2), then give a proof of the correspondence. After that, I’ll mention a few results on small-diameter McKay graphs. I’ll introduce all the necessary pre-requisites to the proof, so anyone with a bit of knowledge of group theory will be able to follow.

**Rufus Lawrence** (Imperial)- Pointless topology and how to fix measure theory

Measure theory (in particular the Lebesgue measure on Euclidean space) plays a fundamental role in many branches of mathematics, particularly in analysis. However, the existence of non-measurable sets gives rise to some strange results, most notably the Banach-Tarski paradox. Motivated by this inconsistency, we develop the theory of (sigma-)locales, which generalise measure spaces and topological spaces respectively. I will conclude presenting one of the main results of Alex Simpson's 2012 paper, and show how this 'resolves' the Banach-Tarski paradox.

**Harry Petyt** (Currently: Bristol. Previous: Warwick)- Introduction to Tropical Geometry

Tropical geometry occupies an unusual spot in mathematics: despite having a fancy name and being a fairly new area of research, its core ideas are very accessible. The aim of the talk is just to see some of the basics, with plenty of examples and pictures to help out.

**Elliot Macneil** (Imperial) -Formalising mathematics with automated theorem provers

Lean is an example of a recently developed "automated theorem prover" - a computer program which can verify that a given statement is a logical consequence of a built-in set of statements, possessed by the program. This gives us the ability to know with much more certainty that a given proof is correct, and spot errors in a proof. One of the distant goals of automated theorem proving, is providing proof verification of any given proof, independent of length or complexity. In this talk, I will cover what specifically Lean is, the mathematical underpinnings of Lean, why theorem provers are important to the future of mathematics, and current developments pertaining to Lean. This talk should be accessible to anyone, and requires no knowledge of Lean, or automated theorem provers to understand the material.

Special thanks to the department for their generous funding and to Rufus Lawrence for organising the Imperial side of WIMP.