# Warwick-Imperial Autumn Meeting

#### 2nd December 2017

Update: day running half an hour late, all talks shifted by this amount! Apologies

Organisers: Ben Windsor (Warwick), George Wynne (Warwick), Shinu Cho (Imperial)

WIMP is a one day conference primarily aimed at 3rd/4th year undergraduate, masters, and early PhD students from Warwick and Imperial though of course others of all ages and locales are welcome, it is also free to attend. We commence with a plenary talk and then divide into two streams of five talks. All talks will be approximately 45 minutes in length with 15 minutes afterward for questions, discussion, and refreshment.

To register for the conference (which is helpful for booking and food organisation for us) fill out this form https://tinyurl.com/WIMP2017Dec

If you have any other questions, please email B.Windsor@warwick.ac.uk or G.Wynne@warwick.ac.uk

**Schedule for the day:**

**09:30:** Informal welcome with tea/coffee (common room).

**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). Once in the building, head up the stairs diagonally to your right. At the top, turn left and you'll see the common room ahead of you, behind a door with glass panels either side.

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**10:00-11:00** Plenary MS.04

**Prof. Keith Ball: **Plank problems and sphere packing

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**11:00-12:00**

[Room MS.04] - Pip Goodman (Bristol): Is Galois theory a lousy lover?

[Room MS.05] - Rufus Lawrence (Imperial): The Cauchy problem for the Einstein field equations

**12:00-13:00**

[Room MS.04] - Esmee Te Winkel (Warwick): Low-dimensional manifolds and their geometry

[Room MS.05] - Trish Gunaratnam (Warwick): The 2D Equilibrium and Non-Equilibrium Ising Model

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**13:00-14:00**: Lunch (common room)

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**14:00-15:00**

[Room MS.04] - Alex Wendland (Warwick): Understanding free groups via graphs

[Room MS.05] - Philip Herbert (Warwick): Particles in Biomembranes

**15:00-16:00**

[Room MS.04] - Álvaro L. Martínez (Imperial): Representation theory of algebraic groups

[Room MS.05] - Tanuj S. Gomez (Imperial): Complex singularities and Milnor fibrations

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**16:00-16:30**: Afternoon tea/coffee (common room)

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**16:30-17:30**

[Room MS.04] - Shinu Cho (Imperial): Brauer groups with Galois cohomology

[Room MS.05] - Tom McDonald (Warwick): Topics in vector measures - the gateway to parabolicity

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**17:30**: Conclusion and conference dinner. Once all is wrapped up we will walk over to the Varsity Pub (https://www.greatukpubs.co.uk/varsitywarwick) to be there for about **18:00.**

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Abstracts:

**- Prof. Keith Ball (Warwick)**: Plank problems and sphere packing

In the early 30s Tarski posed his "plank problem". It was solved very elegantly by T. Bang about 20 years later.

I will explain the solution and show how it can be used to find packings of spheres in high dimensions.

**- Pip Goodman (Bristol)**: Is Galois theory a lousy lover?

We've all noticed it, and frankly, we're not impressed. Group theory has just about put its back out for Galois theory, and what has it done in return? Well, as far as most of us see, just about nothing! But the two were born for each other! So Galois theory must have surely done something for Group theory in return, but what?

On Saturday, we'll have a glimpse at what the two have been getting up to while no one's been looking.

What?! You're still reading this? You mean the above wasn't enough to convince you already? Oh, what's that? You're actually just super interested and just want to know more? Okay well, I suppose I'm feeling generous, so here's a rough idea:

We'll be experiencing two tales about Galois theory and Group theory. The first revolves around a result of Schur, which under a few conditions gives an explicit bound on the order of a finite subgroup of the general linear group. The second will feature Schur indices, and here we'll have a glimpse at their role in a generalisation of the BSD conjecture.

I will assume very little background - Anyone who knows what groups and vector spaces are should be fine. We'll see many examples throughout the talk, which illustrate any definitions that are given.

**- Rufus Lawrence (Imperial)**: The Cauchy problem for the Einstein field equations

Einstein’s theory of general relativity remains one of the most successful physical theories constructed; its predictions have been experimentally confirmed to a high degree of precision, and its insights have lead to new breakthroughs in a plethora of scientific disciplines. However, the equations at the heart of the theory, the Einstein Field Equations, are of interest to pure mathematicians in and of themselves.

This talk will give an overview of my summer research project, which focused on understanding a proof, based on work by Choquet-Bruhat, of the ‘local’ existence of a solution to the Einstein field equations. We begin by defining the tensor quantities needed to write down these equations, and give some examples of solutions. We then turn to the general case, and sketch the derivation of the Einstein constraint equations. This puts us in a position to state a local existence theorem, which we sketch a proof of.

Time permitting, we will end with a discussion of more recent results, amongst them a ‘global’ existence theorem, as well as a so-called “deZornification” result.

**- Esmee Te Winkel (Warwick)**: Low-dimensional manifolds and their geometry

Given a topological manifold, what is the 'best' way to assign a geometry to it? In an attempt to answer this question, we will explore the relation between topological and geometric manifolds of low dimensions. In dimension 2, an invariant called the genus determines what kind of geometry one can put on the manifold. In the generic case, this will be a hyperbolic geometry, and there are uncountably many ways to do so. In contrast, for manifolds of dimension 3, whenever such a hyperbolic geometry exists it is unique! This is a consequence of Thurston's Geometrization conjecture, which was proved in 2003 by Grigori Perelman and which solves one of the Millenium Prize Problems.

**- Trish Gunaratnam (Warwick)**: Trish Gunaratnam (Warwick): The 2D Equilibrium and Non-Equilibrium Ising Model

This talk will be a very basic introduction to the Ising model of equilibrium planar statistical mechanics, and also a non-equilibrium version of it called heat-bath dynamics. The Ising model was devised in 1920 by Lenz and further studied by his student Ising as a basic model of ferromagnetism - the (temperature dependent) phenomenon of certain materials to retain a non-zero magnetisation with no external magnetic field present. Heat-bath dynamics, or Glauber dynamics, is a non-equilibrium Ising model: that is, it equilibrates to the Ising model in some sense. Mathematically speaking, heat-bath dynamics is a continuous-time Markov chain which is reversible with respect to the Ising model.

In the equilibrium setting, we explore phase transition and symmetry breaking away from high temperatures. In the non-equilibrium setting, we focus on the high-temperature regime where there is no phase transition for the equilibrium model, and we address the issue of how fast and how sharp equilibration occurs.

This talk is open to pretty much anyone - it will be a pure math talk, so don't worry if you have no physics background. The pace will be very gentle at the start and I hope to include a fair few pictures!

**- Alex Wendland (Warwick)**: Understanding free groups via graphs

In this talk I will introduce some concepts within Topology, and use them to prove two results to do with free groups, Neilson Schrier theorem (subgroups of free groups are free) and M.Hall theorem (free groups are LERF). These results originally where proved using combinatorial group theory however the proofs within Topology use some really cool ideas which make them considerably shorter. This is work of Baer and Levi (1936) as well as Stallings (1983)

**- Philip Herbert (Warwick)**: Particles in Biomembranes

The shape of a membrane is important in many biological mechanisms, such as trafficking or signal detection.

One wishes to study how constraints affect the shape. This talk aims to introduce the problem of a membrane with embedded proteins and will discuss some of the tools being used to model this problem supposing that any deformations are small.

**- Álvaro L. Martínez (Imperial)**: Representation theory of algebraic groups

The finite groups of Lie type make up, in a sense, 'most' of the finite simple groups. Despite this, they are rarely discussed in undergraduate subjects.

Let k be an algebraically closed field of characteristic p. In order to understand these finite groups, we will be exploring groups of invertible matrices with entries in k, the (linear) algebraic groups. These are of great interest in algebraic geometry, but we will take the algebraic approach and mostly be interested in their representation theory.

Unlike over the complex numbers, there are still some unanswered questions about their representations over k, which in turn provide the representations of their finite analogues. For instance, the dimensions of their irreducible representations are unknown. We will formulate two theorems of Chevalley and Steinberg that constitute the starting point of their study, and we will see some applications of this in finite group theory.

**- Tanuj S. Gomez (Imperial)**: Complex singularities and Milnor fibrations

Picture the curves y = x^3 and y^2 = x^3, and in particular the point at the origin. The latter has a cusp, or a singular point, at the origin. These two curves behave quite differently; if you look at both these pictures in complex variables, the singular case looks twisted and knotted in some sense. Milnor provides a method of studying these types of points with the aptly named Milnor fibrations. These quite nicely provide a way to construct 'exotic spheres', or manifolds that are homeomorphic to the sphere but are not diffeomorphic to its standard smooth structure.

I hope to show you some interplay between various aspects of geometry, namely: algebraic topology, differential topology and perhaps a little knot theory. I shall talk about complex hypersurfaces, fiber bundles, monodromy actions, and time permitting exotic spheres.

**- Shinu Cho (Imperial)**: Brauer groups with Galois cohomology

With a magical combination of group cohomology and Galois descent, one can summarize all the simple algebras over a field using Brauer groups. These groups have a significant role in the foundations of modern class field theory.

We will construct the Brauer group Br(K) of a field K from the classical viewpoint of central simple algebras, and gently borrow ideas from nonabelian cohomology in its description. From this description, we'll see that there are only two central division algebras over the real numbers. We'll then specialize to local fields and examine Hasse invariants, with a peek at a certain local-global principle for algebras over number fields.

The first half of this talk should be fairly accessible (assuming a little Galois theory), but some heavier ideas from algebraic number theory and group cohomology will be used.

**- Tom McDonald (Warwick)**: Topics in vector measures - the gateway to parabolicity

A whirlwind tour of vector measures, Bochner Spaces and a breif discussion on the applications to PDEs. Ever wondered what happens when a measure takes a value in an arbitrary Banach space? Me neither, yet it turns out to not be so far away from what we already know, and has had huge implications on the theories of measures, PDEs and Banach spaces. Only a few measure theory and functional analytic results will be assumed, with the emphasis on ideas not proofs to keep things accessible.

Special thanks to the department for their generous funding and to Shinu Cho (WIMP Imperial Chair) for organising the IMP bit of WIMP.