# Warwick Imperial Autumn Meeting

## Saturday 28th November 2015

###### Organiser: Andrew Brown

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. The day will begin 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 afterwards for questions, discussion, and refreshment.

To register for the conference (which is helpful but unnecessary) or request further details please email **wimp.warwick@gmail.com** and include your home institution in your email, as well as any other relevant data. Food has been booked so new food requirements can no longer be catered for, which is hardly a reason to not attend!

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

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10:00-11:00 Plenary (MS.01)

Prof. Gavin Brown (Warwick), *Möbius strips, flops and flips.*

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**Talks #1
**

11:00-12:00

**MS.04** Dalton Fung (Imperial), *Probabilistic Number Theory*

**MS.05 **Oliver Crook (Warwick), *The Dynamics of an Infectious Disease Near Extinction*

12:00-13:00

**MS.04 **Pip Goodman (Warwick), *Modular Symbols for S-Arithmetic Subgroups*

**MS.05*** *Asbjørn Riseth (Oxford), *Decision-making Under Uncertainty *

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

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**Talks #2
**

14:00-15:00

**MS.04 ** Florian Bouyer (Warwick), *Rational Curves on Special Quartic Surfaces*

**MS.05 ** Anfernee Lo (Imperial), *Nonlinear Wave Equation*

15:00-16:00

**MS.04 **Rosemberg Toala (Warwick), *Introduction to Gravitational Radiation*

**MS.05** Edward Pearce (Warwick), *Finite Special Linear Subgroups and McKay Quivers*

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

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**Talks #3
**

16:30-17:30

**MS.04 **Alexsander Horawa (Imperial), *Primes in Arithmetic Progressions and L-functions*

**MS.05 **Alex Wendland (Warwick), *Planar Groups*

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17:30 Conclusion and conference dinner (The Graduate ⊂ Dirty Duck; campus pub).

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**Titles and abstracts:**

**Prof. Gavin Brown:** *Möbius strips, flops and flips*

With a bit of effort, you can imagine pinching the central axis of a Moebius strip to a point. (Hint: cut along the axis first and then shrink the length of the resulting additional boundary component.) The result is a disc. Writing equations for this contraction map reveals one of the main players in the theory of complex surfaces: Castelnuovo's contraction of a rational curve with normal bundle -1. The same idea connects singularity theory, knot theory and algebraic geometry. I'll show some of its incarnations - some of which you may know better than I do - and then explain the words in the title and how they fit in to parts of modern complex (and other) geometry.

**Dalton Fung:** *Probabilistic Number Theory*

Many would agree that number theory is the study of primes. One of the questions we can ask, is how we can count prime factors of any given (positive) integer n, or in other words, how many prime factors we can expect n to have? Such a phrasing naturally suggests playing around with probability. In this talk we will take a brief glimpse into convoluted interplay between probability and number theory, and establish a few interesting results. There are absolutely no prerequisites to this talk, so it shall be accessible to everyone. Shall time permit, I will also discuss some other general work such as the Cramér's model of the primes (~1936) and Maier's Irregularity Result (~1985).

**Oliver Crook: ***The Dynamics of an Infectious Disease Near Extinction*

There is an international drive to rapidly eliminate a number of neglected tropical diseases in poor populations. Yaws is such a disease, infecting only 2.5 million people, but causing them considerable pain and discomfort. This poorly understood disease mainly affects young children, with an incubation period of between 9 and 90 days. Lack of research and data means that designing effective control policies for this disease is challenging. The project will develop an epidemiological model, using a vast array of mathematical tools, which will be analysed to see how the assumptions in the model lead to the observed dynamics. The questions we seek to answer are what is the impact of earlier diagnosis? How does this compare with mass drug administration? And, how fast could we eradicate the disease? The study will allow us to add to the evidence supporting decisions about the best use of resources in tackling Yaws and under what condition the eradication target of 2020 set by the World Health Organization (WHO) is achievable. The results of the project suggest to us that the WHO’s target is unrealistic and an adapted strategy is needed if eradication by 2020 is to be achievable and cost effective. The data and models also suggest that schools should be the targeted location for treatment. However, our model has limitations and the disease displays complex endemic and epidemic characteristics which are not fully captured by the model and hence more research needs to be done for a definitive conclusion.

**Pip Goodman: ***Modular Symbols for S-Arithmetic Subgroups*

We look at generalising an algorithm originally developed by Darmon-Pollack [DP] which computes special points (Stark-Heenger points) on elliptic curves that are conjectured to be rational. Rational points on elliptic curves are of particular interest as they form a finitely generated abelian group under the addition law. However, they are hard to find using naïve algorithms so mathematical theory is required. In addition to this a better understanding of Stark-Heenger points would lead to progress on important questions in number theory such as the BSD conjecture. We use Shapiro's Lemma to simplify the required homology calculations in Darmon-Pollack's algorithm. Shapiro's Lemma allows us to work with a simpler group at the cost of increasing the coefficient module. This generalisation allows us to compute Stark-Heegner points which were previously unreachable and hence provide more evidence for their conjectured rationality. The collected evidence should hopefully motivate further research into Stark-Heegner points.

**Asbjørn Riseth: ***Decision-making under uncertainty*

Two common issues when making decisions are competing objectives and incomplete information. Demand models are used to motivate these issues in pricing decisions for retailers. We will look at constrained optimisation from a multi-objective viewpoint and how Pareto-fronts may help understand trade-offs between objectives. To handle uncertainty, we give an overview of how risk measures create well-posed optimisation problems.

**Florian Bouyer: ***Rational Curves on Special Quartic Surface*

Arithmetic geometry is a branch of mathematics that tries to answer number theory questions using algebraic geometry. For example, given an equation number theorist try to find rational solutions. In this talk, we start with an equations which describe a surface, and we try to find rational points by answering a different question "does there exists rational conics on this surface?". Along the way, we will highlight how Galois theory and the theory of Elliptic curves help us arrive to a solution.

**Anfernee Lo:** *Nonlinear Wave Equation*** **

Differential equations show up everywhere. Apart from being interested in its own right, it has uses both in pure mathematics to applied mathematics - ranging from studying Riemannian manifolds to effectively everywhere in applied mathematics to model real world phenomena. It turns out - however - that many basic questions about differential equations turn out to be really difficult questions in analysis. Existence and Uniqueness of solutions to differential equations is one of them. In the case of ODEs, Picard’s existence and uniqueness theorem gave us a satisfying answer for a large amount of cases. However, in the case of PDEs this is significantly more difficult. Using the nonlinear wave equation as a model equation, I will talk about some of the tools (in particular, harmonic analysis) used in demonstrating the existence solutions of certain PDEs.

**Rosemberg Toala:** *Introduction to Gravitational Radiation*

In spite of having 100 years old, the theory of General Relativity still has many open questions that remain unsolved. One of uttermost importance is that of finding solutions to the N-body problem, this being an impossible task (think of the Newtonian analogue) one is led to study properties of such solutions. This talk will focus on the "far-away zone"; a linear approximation method and properties of the solutions will be presented. If time permits I will go over what is known about the full non-linear problem and the difficulties that arise. Of course, nothing will be assumed and I will start by recalling definitions and concepts that will take us from Special Relativity and Differential Geometry all the way to hyperbolic PDEs.

**Edward Pearce:** *Finite Special Linear Subgroups and McKay Quivers*** **

The 2D McKay Correspondence is a result which relates algebraic geometry and representation theory, setting up a bijective correspondence between the finite subgroups of SL(2;C) and the Dynkin diagrams An, Dn, E6, E7, E8. The nature of such a relation in the case of three dimensions has been studied thoroughly in the abelian case, and also for particular families of non-abelian subgroups, but a complete uni- fying theory is not yet known. I will introduce concepts and results in representation theory which can encode information about groups and discuss how these can be applied to study the sporadic finite subgroups of SL(3;C).

**Alexsander Horawa:** *Primes in Arithmetic Progressions and L-functions*

The theory of L-functions provides a way to study arithmetic properties of integers (and, more generally, integers in any number field) by first, translating them to analytic properties of certain functions, and then, using the tools and methods of analysis, to study them in this setting. For instance, consider the following statement: Every arithmetic progression whose difference and the first term are coprime contains infinitely many prime numbers. In order to prove it, Dirichlet constructed L-functions, which are given by series resembling Riemann’s zeta function (the simplest example of an L-function) but that may contain some extra coefficients. Using analytic properties of these functions, he showed that the sum of reciprocals of the primes in the progression diverges, and hence there must exist infinitely many of them. I will introduce Dirichlet L-functions and present his proof of the above statement. I will show how we can generalize the function to encode not only properties of rational primes, but also properties of primes of a more general number field.

**Alex Wendland: ***Planar Groups*

Suppose that you have a group, and some topological intuition about it. What does this property have to do with how you can present the group? What does a group that 'looks like' a group with this property have to do with that topological property? Lastly, how nice are the presenations of such groups? This talk will seek to explore these questions in the set up of Plannar Cayley Graphs. I hope the talk will be accessable to all areas, it will have no real hard details just a discussion of the topic with lots of fun pictures!

Special thanks go to Ed Pearce, Laasya Shekaran, Yvonne Collins, Hazel Higgins, Colin Sparrow, Gavin Brown and Nav Patel for all of their help and support.