# WIMP Spring 2019

Saturday 9th 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 1 hour in length, with 15-minute intermissions between talks for questions and refreshments.

# Contact the Organisers:

Rebecca Myers (Warwick) : r.myers.1@warwick.ac.uk

Rufus Lawrence (Imperial) : james.lawrence15@imperial.ac.uk

# Getting to the Event:

The event will be held in the Huxley Building (180 Queen’s Gate, SW7 2AZ). A map of the campus is available here.

# Signing Up

Imperial staff and students interested in attending should sign up here.

Non Warwick or Imperial students should email Rufus Lawrence at the email address above.

# Event Schedule and Talks

# Abstracts

## Ambrus Pal - The absolute Galois group of fields and embedding problems

The absolute Galois group of a field is a group which conveniently packs together all the information about every Galois extension of the given field. It is a central topic of research; for example the Langlands program studies the representation theory of the absolute Galois group of number fields. Another famous work is Voevodsky’s proof of the Milnor and Bloch-Kato conjectures, which computes certain cohomology rings attached to the absolute Galois group in terms of very simple generators and relations. One of the major ways to study the absolute Galois group is via embedding problems. These are not solvable in general, but recently a conjecture was made which identified a large class of embedding problems which should always have a solution. In my lecture I will introduce the absolute Galois group, embedding problems and talk about a special case of my joint work with Tomer Schlank, where we establish cases of the aforementioned conjecture, which can be proved without the use of heavy machinery.

## Raphael Pellegrin - Introduction to Tropical Geometry: Toward a Tropical Nullstellensatz

The set of tropical numbers is defined as R∪{-∞}. We endow this set with two operations, called tropical addition (the usual maximum) and tropical multiplication (the usual addition). The adjective tropical was coined in honour of the Hungarian born Brazilian mathematician Imre Simon (1943-2009).

A basic issue in tropical geometry, a new field of mathematics that emerged at the beginning of the 21st century and based on the initial work of Imre Simon, is to find feasibility and infeasibility certificates for systems of polynomials equalities and inequalities. The simplest situation concerns the case of linear inequalities. It has been shown that decision problems for systems of linear inequalities are equivalent to mean payoff games, a problem whose complexity is unsettled. Recently, Grigoriev and Podolskii, also relying on game techniques, have established a tropical analogue of Hilbert’s Nullstellensatz. They showed that a tropical prevariety, i.e. the intersection of finitely many tropical hypersurfaces, is empty if and only if a certain game, constructed appropriately, is winning. The purpose of this talk is to introduce key notions in tropical mathematics and to mention some tropical analogues of well-known theorems, including the tropical Nullstellensatz of Grigoriev and Podolskii.

## Philippe Michaud-Rodgers - Sums of Three Squares

Every natural number can be expressed as the sum of four integer squares. This is Lagrange’s celebrated four-square theorem, but what about three squares? No matter how hard you try you will find that you cannot write 7 as a sum of three squares! In this talk I will discuss the proof of the three-square theorem. The three-square theorem states that a natural number n can be expressed as a sum of three squares if and only if n is not of the form n=(4^k)(8m+7) for k and m natural numbers. Along the way we will discuss Gauss’ class number problem, classify primes of the form x^2+ny^2 for some small values of n, and show that any natural number can be expressed as the sum of three triangular numbers (known as the Eureka theorem!). The proof is based on the theory of quadratic forms over the integers, a very rich area of mathematics and an active area of research. The talk will be accessible to all, the only prerequisite being knowledge of modular arithmetic.

## Gautam Chaudhuri - What we mean by the Lagrangian: Differential Geometry in Mechanics

Many of us are familiar with the Newtonian formalism of mechanics from high school and some early undergraduate modules. It is one of the simplest mechanical formalisms, but this simplicity comes at the cost of having to introduce complex methods to solve certain mechanical systems. These methods may also not work for systems in general and so the entire framework relies on a ‘divide and conquer’ strategy. The benefit of the Lagrangian formalism is that it relies on properties intrinsic to the system to describe it, this can lead to results that apply to any system described by this framework.

In this talk I will introduce the Lagrangian formalism as an alternate way to solve mechanical problems and discuss one of its extensions. I will begin by defining the formalism through a (semi-)rigorous geometric foundation based on classical particles. This will then be extended to cover a more general setting which can be viewed as a very basic precursor to Yang-Mills Theory. Finally, I will state Noether’s theorem in the context discussed up until now. This talk assumes familiarity with differential geometry, particularly with connections and Lie groups.

## James Taylor - The Many Faces of Matroids

A matroid is a combinatorial generalisation of the notion of linear independence of vectors, which can be applied to many areas of mathematics, including graph theory. This talk will be an introduction to the theory of matroids, surveying many of the cryptomorphic (equivalent, but not obviously equivalent!) characterisations, including the most surprising definition as ’systems’ for which the ‘greedy algorithm’ provides an optimum solution. We will see how far we can generalise many of the useful notions we are used to from linear algebra, such as rank, span and bases, and take a look at some big questions in this area, finishing with some recent results resolving Rota’s conjecture. The only prerequisites required is elementary linear algebra, so this talk should be accessible to all!

## Ali Barkhordarian - Moduli Spaces 2: Algebraic Boogaloo

*One cop, in a city pushed to the edge. Out for revenge, his wife and daughter lost. Join us for the sci-fi action thriller of the year, as well as a discussion of the Grassmanian in algebraic geometry leading to thinking about moduli problems. This is meant as an introduction to algebraic geometry. Starring Cameron Mitchell and David Caradine, America is not ready.*

## Andrew Darlington - The Powerful Theory of Modular Forms

Truly beautiful mathematics is created when two seemingly unrelated fields are suddenly brought together for the first time in history. It allows us to use developed techniques from both fields to drive the newly found intersection to produce very powerful tools and objects capable of shedding light on many existing problems, developing an entirely new field in the process. Modular forms bring together the areas of complex analysis and number theory, and have enabled mathematicians to prove some of the most important results in mathematics. The Taniyama Shimura conjecture – more colloquially known as the “modularity theorem” (stating that elliptic curves are in some way related to modular forms) was instrumental in proving the great Fermat’s Last Theorem. They are also closely related to the Riemann zeta function, and thus of course the famous Riemann hypothesis finds its home in the field of analytic number theory (the field in which modular forms reside). In this talk, I will introduce modular forms, give some of the initial theory, and talk about how they can be used to solve problems such as counting the number of ways of writing integers as the sum of an even number of squares. I will end by talking about extensions to the theory - focussing mainly on modular forms of ½ integral weight, and how they, under certain conditions, can be related back to the much better understood world of integer weight modular forms. I will assume basic knowledge of algebra and analysis, so this talk should be accessible to all!

## Tanuj Gomez - Brief encounters of the (-1)-kind: Castelnuovo’s contraction criterion

In the crudest of terms - Algebraic Geometry is the study of polynomials and the ‘varieties’ cut out by them. One of the most fundamental questions we can is can we classify these varieties? Well for algebraic curves, we can classify them with invariants such as the genus. For higher dimensions this unfortunately blows up! I will say something about how we blow up and how to salvage the wreckage through Castelnuovo’s Contraction Criterion. Time permiting I will say some things about the Minimal Models Programme and The Kodira-Enriques classification for surfaces.

## Lawk Mineh - Relatively Hyperbolic Groups

Hyperbolic groups play a central role in geometric group theory. Broadly speaking, they model fundamental groups of compact hyperbolic manifolds. Relatively hyperbolic groups form a natural extension of this theory, approximating the fundamental group of a noncompact (but finite volume) hyperbolic manifold. Examples are manifolds with ‘cusps’, or points at infinity. There is a natural way we can assign a topological boundary to hyperbolic and relatively hyperbolic groups. In this talk we will explore how, in particular, connectedness properties of the boundary correspond to splittings of groups into smaller algebraic chunks. A little bit of topology and group theory will be helpful to understand this talk. Familiarity with some hyperbolic geometry will be useful too, but we will quickly revise this at the start.

## Gabriel Ng - Morley’s Categoricity Theorem: Classical results in Model Theory

Model theory is, generally speaking, the study of the relationship between first-order theories in formal languages and the structures that satisfy these theories. In particular, one can ask when a first order theory will define a structure uniquely up to isomorphism. Unfortunately, by the Löwenheim-Skolem theorem, the answer to this is almost never, unless it defines a finite structure in a finite language.

In this talk, we will look at the model-theoretic notion of categoricity, which restricts our attention to fixed cardinalities. We will see a few accessible examples of countably and uncountably categorical theories, and look at some connections between categoricity and other notions in model theory.

## James Hefford - Categorical Quantum Mechanics: Why Diagrams Rule!

Categorical quantum mechanics is a very recent area of research started by Abramsky and Coecke in 2004. It concerns itself with using category theory to formalise the structures of quantum mechanics at a higher-level. This new language allows one to construct a beautiful graphical calculus which can be used to both prove results and perform calculations.

In this talk the basics of category theory will be introduced as well as the required quantum mechanics. In particular we will see that the no-cloning theorem is in fact a more general result on braided monoidal categories. There should a bit for everyone in this. Category theory for the pure folk, quantum mechanics for the applied folk and snakes to bring everyone together. Indeed, if you like snakes you are in for a treat!