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Warwick Imperial Autumn Meeting

Saturday 29th November 2014

Organiser: Ben Wormleighton

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; focusing on algebra, geometry, and number theory and analysis, discrete maths, and topology respectively. 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 (dietary needs, etc).

9:30 Informal welcome with tea/coffee/infusions (common room)

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

Galois module structures, Dr Alex Bartel (Warwick)

Talks #1 MS.04 (algebra, geometry, number theory) MS.05 (analysis, discrete maths, topology)
11:00-12:00 Alex Best (Cambridge)

Singular moduli

Paul Druce (Warwick)

Algebraic knots in liquid crystals

12:00-13:00 Ben Wormleighton (Warwick)

Mirror symmetry through toric geometry

Alex Wendland (Warwick)

The four colour theorem and its extensions

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

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Talks #2 MS.04 (algebra, geometry, number theory) MS.05 (analysis, discrete maths, topology)
14:00-15:00 Dominic Bunnett (Warwick)

William in the sky with diamonds

Louis Bonthrone (Warwick)

Censoring singularities in general relativity

15:00-16:00 Clara Dolfen (Imperial)

Graph colouring via Groebner bases

Tasmin Symons (Imperial)

Random matrices and the Riemann hypothesis

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

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Talks #3 MS.04 (algebra, geometry, number theory) MS.05 (analysis, discrete maths, topology)
16:30-17:30 Andrea Dotto (Cambridge)

The adjoint representation

Tim Westwood (Imperial)

The existence of Fatou coordinates

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

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Titles and abstracts:

Alex Bartel: Galois module structures

The rich and versatile branch of number theory that the title of this talk refers to aims to understand how Galois groups act on arithmetic structures. If K/Q is a Galois extension, then almost any natural invariant of K comes with a Galois action, be it the ring of integers, or its unit group, or the class group of K, you name it. Trying to understand the structure of these objects not just as groups, but as Galois modules leads to surprising connections within number theory and to other fields of pure mathematics. This area has experienced dramatic advances in the last few decades, but lots of questions remain open. Virtually no number theoretic background will be assumed, except for basic Galois theory.

Alex Best: Singular moduli

Singular moduli is the name given to certain special values of a very special function (the j-invariant). These values were studied by number theorists in the 19th century and many thought the story was complete, until more new properties of these values were discovered in the early 80's. In this talk we'll look initially at the classical theory of singular moduli, how they relate to class field theory and how we can use them to explain some striking arithmetic equalities. After this I'll try to provide an overview of some more recent work concerning singular moduli such as Gross and Zagier's work on factoring their differences.

Paul Druce: Algebraic knots in liquid crystals

In recent years, it has been made possible to create arbitrarily knotted liquid crystals. These are liquid crystals that have knotted line defects within them. Theoretical models of such a system are difficult to deal with, due to the complex nature of globally knotted continuous fields. This talk will be a description of how Milnor's Fibration Theorem and elements of algebraic geometry are being used to tackle this problem.

Ben Wormleighton: Mirror symmetry through toric geometry

A routine strategy in algebraic geometry is to reduce interesting geometrical objects to a packet of combinatorial data. This is epitomised by the aesthetic theory of toric varieties in which key activities such as resolution of singularities or computation of invariants become largely pictoral. I will introduce the basic setup for toric geometry – in doing so introduce orbifold singularities – and continue on to its appearance in mirror symmetry. After its tenuous origins in string theory, mirror symmetry has united research programmes at Imperial and elsewhere with its startling conjectures and results in previously inaccessible parts of algebraic geometry. I will provide a gentle introduction to some aspects of the programme in the sense pioneered at Imperial.

Alex Wendland: The four colour theorem and its extensions

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colours. Fabrici and Goring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1,..,4 in such a way that every face contains a unique vertex coloured with the maximal colour appearing on that face. The talk will consist of current work on this problem as well as a conjectured list colouring variant.

Dominic Bunnett: William in the sky with diamonds

In the 1930s some of the most interesting consequences of William Hodge's work on De Rham cohomology were coming to fruition. Arguably the most interesting of these consequences is the splitting of the cohomology groups H^k(M) into H^{p,q}(M) . Hodge Diamonds are a result of some very nice symmetries within the dimension of these groups, and it is these that we will begin investigate.

Louis Bonthrone: Censoring singularities in general relativity

We give a brief overview of the mathematical structure of general relativity, introduce the Penrose diagram representation and provide a definition of singularities in our context. Next the Penrose Singularity Theorem is stated and discussed, but only in the generality which we will need it. We then move to look at the Schwarzschild and Kerr space times, which will motivate the Strong Cosmic Censorship Conjecture.​ This conjecture is an area of current research. For completeness the Weak Cosmic Censorship Conjecture will be given but with only brief discussion.

Clara Dolfen: Graph colouring via Groebner bases

There are several ways to determine the existence of a proper k-colouring of an undirected graph. One of those methods uses Groebner bases and relies on encoding the structure of the graph in form of polynomials. Once the structure is encoded, the hard work is done and we are left to find solutions to the system. So if we are thrown any graph, we can determine existence and even get an explicit colouring. But can we say more? As it turns out the structure of the reduced Groebner bases associated to the graph gives away a lot of information, some of which I will discuss.

Tasmin Symons: Random matrices and the Riemann hypothesis

This talk explores the well-known but conjectural link between the Riemann Hypothesis and nuclear physics. First posed by George Polya in correspondence with Edmund Landau a century ago the conjecture - that the zeros of the Riemann Zeta Function correspond to the eigenvalues of a Hermitian operator more often seen in quantum mechanics - lay dormant until a fortuitous tea-time meeting between Hugh Montgomery and Freeman Dyson in 1972.

In the proceeding forty years the field has exploded into an active and exciting field of research, with every new paper offering increasingly persuasive support of Polya's conjecture. We paint a broad picture of the theory, in a manner accessible to all years and mathematical backgrounds.

Andrea Dotto: The adjoint representation

If we are to go by the claim that tangent spaces are local linear models to a manifold, we should expect to see something algebraic going on in the tangent space to the identity of a Lie group. Indeed, this naturally carries a bilinear form with two extra properties--the resulting structure is known as a Lie algebra, and it could be approached in a purely algebraic way. In this talk, I'll try to go slightly deeper into the meaning of the word "naturally" in this context. Unsurprisingly, this will involve a form of functoriality.

The adjoint representation is a smooth representation of the group on its tangent space at the identity, and its differential encodes the Lie bracket. We'll see how the Lie algebra axioms, perhaps not the first thing you'd think of when meeting a bilinear form on your way home, are a direct consequence of the functoriality of these constructions with respect to Lie group homomorphisms and their differentials.

Tim Westwood: The existence of Fatou coordinates

Near a parabolic fixed point there is a beautiful description of the dynamics known as the Leau- Fatou flower theorem, where attracting and repelling axes alternately surround the fixed point forming petals. There exists a coordinate transformation, the so called “Fatou Coordinates”, where the behaviour of points in these petals can be conjugated to “translation by one”. But what about other points close to these petals? I aim to discuss an extension of these Fatou Coordinates to the maximal domain of definition, the filled Julia set.

This is both a satisfying and visually appealing result in Complex Dynamics and this talk should be accessible to (almost) anyone!