# Colloquia Abstracts 07/08 Spring Term

Friday 11 January 2008 **Mary Rees** (Liverpool) *Counting Maps*

The motivation for this colloquium talk is the study of some very specific maps between specific finite sets arising in a parameter space of complex dynamical systems. In general one would not expect a given map between finite sets to behave in a "typical" fashion. But typical behaviour of some sort might be expected if a map satisfies some sort of recursion to a high level. I shall discuss relevant example given by an ideal polygon fundamental domain for a free discrete group of hyperbolic isometries acting on the hyperbolic plane, and a finite set of geodesic rays from a basepoint in F to translates of the vertices of F.

We shall consider some very basic questions about maps between finite sets, such as:

What is the average image size of a map from a set of m elements to a set of n elements?

What about average maximal inverse image size?

Then I shall discuss how far the specific maps diverge from the average.

Friday 18 January 2008 **Tim Dokchitser** (Cambridge) *Parity conjecture for elliptic curves*

This is joint work with Vladimir Dokchitser. For an elliptic curve defined over the rationals there are various "modulo 2" versions of the Birch-Swinnerton-Dyer Conjecture, each sometimes called the Parity Conjecture. I will explain the ideas behind our recent proof of one of these conjectures (this completes earlier work by Birch-Stephens, Greenberg-Guo, Monsky, Nekovar and Kim) and some of the group- and representation-theoretic questions that arise from them.

Friday 25 January 2008 **Michael Stoll** (Jacobs)* Rational points on curves of genus *

A curve of genus 2 is given by an equation

y^2 = f_6 x^6 + f_5 x^5 + ... + f_1 x + f_0

where the polynomial on the right is squarefree and f_5 and f_6 are not both zero. We will consider curves defined over the rational numbers; we can then take the coefficients to be integers. A rational point on this curve is a pair of rational numbers (x, y) satisfying the equation; in addition there can be rational points "at infinity'' corresponding to the square roots of f_6, if this coefficient is a square.

We will discuss several questions related to rational points on curves of genus 2, for example:

1 Can we decide if there are any rational points on a given curve?

2 Can we determine the (finite) set of rational points on a given curve?

3 How many and how large points can we expect on average?

Regarding this last question, we will present some heuristic considerations and compare them with experimental data obtained from curves with small coefficients.

Friday 1 February 2008 **Raphael Rouquier** (Oxford) *Dunkl operators, microlocalization and quantization*

We introduce certain algebras of deformed differential operators on a vector space. Their representation theory can be studied via monodromy representations, leading to Hecke algebras. On the other hand, these algebras can be microlocalized. This microlocalization provides a quantization of the Hilbert schemes of points on the complex plane.

Friday 15 February 2008 **Fabrizio Catanese** (Bayreuth) *A packing problem for automorphisms of Riemann surfaces and the slope of Kodaira fibrations.***
**Kodaira fibrations are differentiable fibre bundles, and were the first counterexamples to the multiplicativity of the signature. The slope lies here between 2 and 3 and measures the failure of this multiplicativity. Kodaira's examples have slope 7/3. In joint work with Soenke Rollenske we constructed new examples with slope 8/3, using algebraic curves with many automorphisms. I will discuss some conjectures and questions, and in the end I will describe some result concerning the moduli spaces of these surfaces.

Friday 22 February 2008 **Richard Kaye** (Birmingham) *Nonstandard symmetric groups*

Interesting "nonstandard symmetric groups" were discovered in joint work with John Allsup, in a recent paper in the Archive for Mathematical Logic. The construction goes by considering the internal finite symmetric groups S = S_n and A = A_n inside a nonstandard model M of arithmetic, where n is nonstandard. The (external) normal subgroup structure of S and A was determined in the paper and it turns out that there is a proper maximal normal subgroup N of A. Moreover, there is a natural metric structure on A/N making this a topological group, and if M is sufficiently saturated this metric space structure is complete.

This talk will review this construction and then outline some of the further (external) properties of these "nonstandard symmetric groups" that have been discovered since the publication of the paper. In particular A/N is not locally compact but has an interesting measure structure that will be discussed.

Friday 29 February 2008 **Mark Roberts** (Surrey) *The geometry of spacecraft attitude control*

Conservation of angular momentum enables the attitude of a spacecraft to be controlled by "momentum exchange" devices such as momentum wheels and control moment gyroscopes (CMGs). However the mapping from actuator parameters, such as momentum wheel speeds and CMG gimbal angles, to rotations of the spacecraft is highly nonlinear and suffers from the resence of singularities. Understanding the nature and consequences of these singularities is of fundamental importance in the design of control algorithms. In this talk I will give some examples of this, drawing on joint work with colleagues in the Surrey Space Centre and Department of Mathematics.

Friday 7 March 2008 **Anton Thalmaier **(Luxemburg) *Brownian motion of Jordan curves and stochastic analysis on the diffeomorphism group of the circle*

We start with differential geometry of the diffeomorphism group of the circle and explain how Brownian motion on the space of Jordan curves is constructed by solving a "welding" problem of "sewing" together conformally the interior and exterior of the unit circle, glued on the unit circle by diffeomorphisms. Using Kirillov's point of view, our approach leads to stochastic analysis on the space of univalent functions on the complex disk. This is joint work with H. Airault and P. Malliavin and part of a project to construct unitarizing measures for representations of the Virasoro algebra.

Friday 14 March 2008 **Dorothy Buck **(Imperial) *Predicting DNA Knot/Link Type after Protein Action*DNA molecules often have a circular, or topologically constrained, central axis. The topology of this axis can influence which, and how, proteins interact with the underlying DNA. Subsequently, there are protein families (e.g. recombinases) that change the DNA axis topology, for example converting an unknot into a torus knot.

Experimentally determining the minimal crossing number (MCN) of the newly formed knots or links is tractable, but determining the exact knot/link type is very difficult and expensive. Unfortunately there are there are 1,701,936 knots with MCN < 17, so a finer sieve for predicting the knots/links that arise from protein action is needed.

In this informal talk, I'll discuss some recent work that proves that all knots/links formed during DNA recombination must fall within a single well-defined family. I'll discuss what knowing the knot or link type tells us about the biological mechanism of recombination. And I'll conclude by showing how this work can now determine previously uncharacterized experimental results.

(No prior biological knowledge necessary.)References:

- A topological characterization of knots and links arising from site-specific recombination (D Buck and E Flapan) J Phys A (2007) doi:10.1088/1751-8113/40/41/008
- Predicting Knot or Catenane Type of Site-Specific Recombination Products (D Buck and E Flapan) J Mol Biol (2007) doi:10.1016/j.jmb.2007.10.016
- Classification of Tangle Solutions for Integrases, A Protein Family that Changes DNA Topology (D Buck and C Verjovsky Marcotte) J Knot Theory Ram (2007) arXiv: math/0501173
- Tangle solutions for a family of DNA-rearranging proteins (D Buck and C Verjovsky Marcotte) Math Proc Camb Phil Soc (2006) doi:10.1017/S0305004105008431

Mathematics Research Centre

University of Warwick

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