# Warwick Imperial Autumn Meeting

#### 3rd December 2016

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, it is also free to attend. 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 afterward for questions, discussion, and refreshment.

To register for the conference (which is helpful but unnecessary) fill out this form https://docs.google.com/forms/d/e/1FAIpQLSe-FJUaIlr0DlVIRFeE2smzVTiH3Xp6bRPin8iqvBmcoQ9b3A/viewform. If you have any other questions, please email wimp.warwick@gmail.com.

**Schedule for the day:**

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

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10:00-11:00 Plenary (MS.03)

**Dr Adam Epstein (Warwick): Transversality Principles in Holomorphic Dynamics**

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Talks #1

11:00-12:00

MS.03 - **Trish Gunaratnam (Warwick): Stochastically Perturbed Mean Curvature Flow**

MS.05 - **Guhan Harikumar (Warwick): Harmonic Maass forms and Mock Modular Forms**

12:00-13:00

MS.03 - **Asad Chaudhary (Imperial): Hyperbolic Geometry and Complex Analysis**

MS.05 - **Madhavan Venkatesh (Warwick): Stars and Jets**

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

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Talks #2

14:00-15:00

MS.03 -** Matthew Rigby (Oxford): Shock Waves for Hyperbolic Conservation Laws, and application to Shock Diffraction by a Wedge**

MS.05 - **Bradley Doyle: (Imperial): Morita equivalence**

15:00-16:00

MS.03 - **Matthew Schrecker (Oxford): Euler equations: problems with existence, uniqueness and relativity**

MS.05 - **Haiping Yang (Imperial): The CDE Triangle**

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

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Talks #3

16:30-17:30

MS.03 - **Adelina Manzateanu (Bristol): Invariants of Curves of Small Genus**

MS.05 -** Rory Ainslie (Imperial): The virtuous property of the RGB structure**

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

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**Titles and abstracts:**

**Dr. Adam Epstein (Warwick):** Infinitesimal computations in arithmetic dynamics

Cohomology provides a framework for inﬁnitesimal deformation theory in analytic and algebraic geometry. We consider endomorphisms of the projective line over an algebraically closed ﬁeld. Congenial formalism available for arbitrary endomorphisms in characteristic 0 remains available for tamely ramiﬁed endomorphisms in positive characteristic. We pay particular attention to the relations among cycle multipliers, and to the rigidity of postcritically ﬁnite endomorphisms.

**Matthew Schrecker (Oxford):** Euler equations: problems with existence, uniqueness and relativity

The compressible Euler equations of fluid dynamics were formulated by Leonhard Euler in 1757, but remain an active area of research more than 250 years later. In this talk, I will explain the setting of the problem and discuss some of the issues surrounding existence and uniqueness of solutions to the equations before describing the relevance of their study to the study of Einstein's equations in general relativity.

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**Madhavan Venkatesh (Warwick):** Stars and Jets

Algebraic geometry is roughly the study of solution sets to polynomial equations. One wishes to understand more about these things by studying maps between them. Some notions include singularities, points where a variety fails to be a manifold; rationality, when the function field k(X) is isomorphic to a field of rational functions; and that of moduli, parameter spaces for objects satisfying some property. I will introduce a generalisation of the Gauss map for varieties and discuss potential applications to singularities, stable rationality (or ‘starationality’) and moduli.

**Matthew Rigby (Oxford):** Shock Waves for Hyperbolic Conservation Laws, and application to Shock Diffraction by a Wedge.

Hyperbolic conservation laws are used to model many situations in Physics. Firstly, I will introduce the basic theory of hyperbolic conservation laws - discussing the concepts of characteristic curves, shock waves, the Rankine-Hugoniot condition and entropy solutions. Then, I will discuss my research, which deals with shock diffraction by a wedge for the nonlinear wave system, which is a simplification of the full Euler equations for a compressible isentropic fluid, by viewing the shock wave as a free boundary.

**Adelina Manzateanu (Bristol):** Invariants of Curves of Small Genus

This talk will focus on elliptic curves and curves of genus 2. I will explain the importance of invariants and class polynomials, and more precisely I will describe the complex multiplication (CM) method which is one of the most efficient ways of construction hyperelliptic curves defined over large prime fields and suitable for cryptography. Moreover, I will give some details on the difficulties arising in the case of genus 3 curves

**Trish Gunaratnam (Warwick):** Stochastically Perturbed Mean Curvature Flow

In 1992 Evans, Soner and Souganidis showed that in the sharp interface limit (epsilon tending to zero), solutions to the Allen-Cahn equation converge to either +1 or -1 and the boundary dynamic evolves via a mean curvature flow. A similar result was obtained more recently in the case that the Allen-Cahn equation has a smooth time-dependent noise term with the boundary dynamic evolving via a stochastically perturbed mean curvature flow. In this talk I hope to give an overview of this result and some of the techniques used. Time permitting, I would like to briefly discuss the difficulties of replacing a smooth time-dependent noise term with white noise. This talk should be accessible to most 3rd and 4th year students.

**Guhan Harikumar (Warwick):** Harmonic Maass forms and Mock Modular Forms.

In this expository talk, I will introduce the objects in the title and we will look at certain number theoretic and combinatorial results that can be deduced from the theory of Harmonic Maass forms and mock modular forms.

**Haiping Yang (Imperial): **The CDE Triangle

The representation theory of finite groups over an algebraically closed field k of characteristic 0 is generally well understood. In this case, we have Maschke’s theorem, which tells us that every module can be written as a direct sum of simple modules. But when the characteristic is positive (modular representation), Maschke’s theorem can fail, and this happens exactly when char (K) ||G|. So what do we do in this situation? Well, if a module does not split into simple modules, then we just construct a new object where it has to split. The general tool that we will use to analyse this new object is called the CDE triangle. From here on, we can even construct characters for modular representations, and relate them to ordinary characters.

**Asad Chaudhary (Imperial): **Hyperbolic Geometry and Complex Analysis

By introducing the notion of a conformal metric, we can study complex analysis through the lens of geometry. In this talk I will consider a special class of conformal metrics, the so-called hyperbolic metrics on domains in the complex plane. After developing some geometric ideas we will apply them to prove Picard's Big Theorem, a well known (though perhaps surprising) result about the range of a complex function with an essential singularity.

**Bradley Doyle: (Imperial):** Morita equivalence

I will aim to give an overview of Morita equivalence, which is a weakening of isomorphism for rings. To do this will require parts of ring theory, module theory and category theory which I will introduce. Anyone who has seen modules before should be able to get the general ideas.

**Rory Ainslie (Imperial): **The virtuous property of the RGB structure

Homogeneity is a highly useful and therefore much desired property in structures. However, as is often the case, it’s also very rare. In this lecture I will be reviewing some fundamentals of homogeneity on infinite structures, covering some basic results and outlining some new mechanisms for the identification of the property in the case of the RGB structure.

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