# WIMP Spring 2018

Saturday 10th March Imperial College London# What is WIMP?

WIMP is a one-day conference primarily aimed at 3rd/4th year undergraduates and early graduate students from Warwick and Imperial. The event is open to all and is free to attend.

After a plenary talk by a guest speaker, we have organized ten student speakers to speak about their mathematical work. All talks are 45 minutes in length, with 15-minute intermissions between talks for questions and refreshments.

** Please fill in this quick form if you wish to attend. **

# Contact the Organisers:

George Wynne (Warwick): g.wynne@warwick.ac.uk

Shinu Cho (Imperial): shinu.cho15@imperial.ac.uk

# Getting to the event:

**ACE Extension 250 (LT1) and 203 (LT2)**

Campus map

# Event Schedule and talks:

PDF availiable here

**9:30 – 10:00**

Welcome reception with refreshments

**10:00 – 11:00**

Prof. Kevin Buzzard: How not to make a perfectoid space

ACEX 250 (LT1)

Perfectoid spaces are a relatively new trendy thing, which Peter Scholze used to prove new cases of the Langlands conjectures and the weight monodromy conjecture (whatever that means). What is a perfectoid space? It's a kind of adic space. What's an adic space? That's a good question. I'll explain, with some examples. Then I'll go on to say something about my 2015 paper with Alain Verberkmoes about affinoid adic spaces, and finally I'll explain why we're all going to be replaced by computers.

**11:00 – 12:00**

Asad Chaudhary: Partial regularity for harmonic maps into spheres

ACEX 250 (LT1)

It is well known that harmonic functions are smooth. The notion of a harmonic function can
be extended to that of harmonic maps between Riemannian manifolds in a natural way – to what extent do the properties of these functions carry over into the more general setting? Are harmonic maps between manifolds still smooth? In this talk I will present a result of LC Evans establishing that harmonic maps into spheres are smooth except for possibly on a set of low dimension (codimension 2).

Joe Scull: Seifert fibred spaces – the nicest 3-manifolds?

ACEX 203 (LT2)

The Classification Theorem for Surfaces provides us with a comprehensive understanding of 2-manifold topology. In the world of 3-manifolds however, even seemingly simple questions like the Poincaré conjecture went a century unanswered. This gap in difficulty makes 3-manifold topology a vibrant area of research, but also an intimidating one. Seifert Fibred spaces bridge that gap and provide a kinder introduction to the world of 3-manifolds. In this talk I will introduce this class of 3-manifolds and demonstrate how their links with 2-manifolds allow us to better understand them.

**12:00 – 13:00**

Quintin Luong: Mandelbrot set with some mates

ACEX 250 (LT1)

Polynomial mating is a general phenomenon observed first by Douady and Hubbard where the Julia set of a rational map, which contains the chaotic behaviour of a complex dynamical system, can be viewed as the result of gluing the simpler Julia sets of two polynomial maps.

I hope to introduce the concept through examples and in the case of quadratic matings, which will naturally lead us to discuss the Mandelbrot set.

Familiarity with complex analysis and the Riemann sphere will make the talk more digestible. Nonetheless I promise plenty of pretty pictures, and animations even cooler than zooming in on Mandelbrot sets. Bring your own acid.

Harry Petyt: Octonions and exceptional groups

ACEX 203 (LT2)

The aim of this talk is to describe a link between two seemingly unrelated classifications of nice algebraic objects. On the one hand we have the octonions, the big brother of the complex numbers. On the other we have the simple Lie groups, which sit at the meeting point of several areas of mathematics, including algebra, mathematical physics, and PDE theory. Both objects will be introduced, and we will see that in some sense the exceptional simple Lie groups exist because of the octonions.

**13:00 – 14:00**

Lunch

**14:00 – 15:00**

Raphaël Pellegrin: Geodesics on metric connections

ACEX 250 (LT1)

This talk will be a basic introduction to geodesics and should be very accessible. The history of geodesics lines dates back to Johann Bernoulli, who first studied length- minimizers on a convex surface in 1697. After Riemann introduced the notions of Riemannian geometry and Riemannian manifolds, the idea of geodesics had to be generalized. This is motivated partly by the intrinsic geometry of surfaces as well as the occurrence of spaces which cannot be embedded as hypersurfaces in the Euclidean space. This need of explaining all geometric quantities in a totally intrinsic manner, without making references to an ambient space, lead to a new approach in the study of geodesics, where one expresses the geodesic equation with respect to a connection. Using the usual Euclidean metric, we can consider geodesics with respect to different metric connections, such as the Levi-Civita connection - the torsion-free metric connection - as well as metric connections with torsion. Surprisingly, the geodesics of metric connections with torsion have received little attention since the death of Élie Cartan and can qualitatively very different from Levi-Civita geodesics. I hope to show examples of how one can compute geodesics of metric connections with torsion and would like to highlight some of the differences between different families of geodesics.

Aaron Zolnai-Lucas: The discrete Kakeya conjectures

ACEX 203 (LT2)

What's the smallest set you could invert a needle in? Defining such sets (known as Kakeya sets) will be the starting point of this talk, along with stating the conjecture about their fractal dimension. I will cover some of the attempts to "discretize" the problem including a proof of the finite field Kakeya conjecture, and graph theoretic methods used by Katz & Tao to find the current best bounds for dimension. If time permits, I'll also present a proof for why such methods won't be enough to decide the conjecture. Why do we care about these sets? Well, aside from applications in harmonic PDEs, which I won't cover, this topic establishes a beautiful link between analysis and combinatorics, "discrete analysis" if you like. I won't assume much knowledge for this talk, but some measure theory and fractal dimension knowledge will come in handy.

**15:00 – 16:00**

James Hefford: Vortices in Bose-Einstein condensates

ACEX 250 (LT1)

Bose-Einstein Condensates (BECs) are a fairly recent area of mathematical research. Initially predicted by Bose in 1924 and developed further by Einstein in the years following, this strange state of matter remained elusive until 1995 when the first condensate was formed from super-cooled rubidium atoms. In this talk we will discuss some of the mathematical theory of scalar condensates, in particular the Gross-Pitaevskii equation and its implications for the formation of vortices in condensates. Time permitting, the basic ideas of spinors will be introduced and their use in describing condensates with spin degrees of freedom.

While a good understanding of quantum mechanics is needed to grasp the intricacies, enough details of basic quantum theory will be explained that everyone should be able to follow along.

Bradley Doyle: A brief intro to deformation theory via polynomials and Hochschild cohomology

ACEX 203 (LT2)

I will motivate deformation theory and then briefly cover Hochschild (co)homology and its relation to deformations. I will show examples and time permitting, maybe mention the categorical way of thinking about deformations. The examples and motivation should be accessible to everyone, having a basic knowledge of homological algebra, or at least having seen examples of (co)homology will help with the middle part of the talk, but isn't absolutely necessary.

**16:00 – 16:30**

Afternoon refreshments

**16:30 – 17:30**

Jeremy Wu: Inverse problems

ACEX 250 (LT1)

Inverse problems appear ubiquitously in many important areas such as biology, image analysis, and statistics. Given some measurement or observation of the real world (effect), can we find the driving force behind it (cause)? In the classical theory of Inverse Problems, this is translated as finding u, an element of a Hilbert space, such that Ku = f, where K is a given bounded linear operator and f is a given element of a (possibly different) Hilbert space. Natural difficulties that arise are: Do solutions exist (what if f is not in the range of K but perhaps is in its closure)? Can we find 'best' approximations if exact solutions don't exist (least squares)? If we have imperfect measurements, will our recovery process be continuous? Time permitting, most of these questions will be examined under the functional analytic framework of Inverse Problems. 2nd year students at Imperial may find familiarities with the Numerical Analysis course which restricts to finite dimensional Inverse Problems

George Wynne: Ergodic averages of free groups

ACEX 203 (LT2)

My talk will be based on the use of Markov operators to show convergence of ergodic averages of free groups. A clever trick is used to turn the sum over an index depending on n into a finite sum of powers of operators. Along the way the generators of a free group are modelled as states in a Markov process and then martingale convergence theorems are used to gain convergence results. I will not be going into too much analytic detail in this talk, but some basic measure theory is required e.g. familiarity with sigma algebras, conditional expectation etc.

**17:30**

Event conclusion