Seminar Programme - Autumn 2005

All meetings will take place at 4.00pm Thursdays in Lecture Room A on the ground floor of the Watson Building, unless otherwise stated. There will be tea in the common room at 3.30pm. Visitors are welcome.

For further information please contact Dr Daniela Kühn (kuehn@maths.bham.ac.uk)

October 6th: Tony Carbery (Edinburgh) The Brascamp--Lieb inequalities: finiteness, structure and extremals

October 13th: Rob Curtis (Birmingham) The Leech lattice $\Lambda$ and the Conway group $\cdot O$ revisited

October 19th: Group Theory and Applications, LMS meeting

October 20th: Christian Elsholtz (Royal Holloway) Additive decompositions of the set of primes and new methods in prime number theory

October 27th: no seminar

November 3rd: Peter Cameron (Queen Mary University of London) Counting orbits on colourings, flows, and other things.

November 10th: Chris Good (Birmingham) Inhomogeneities in inverse limits of dynamical systems

November 17th: Raphael Rouquier (Leeds) Dunkl operators and Hecke algebras 

November 24th: Andras Zsak (Cambridge) Embedding asymptotically \ell_p Banach spaces into 
asymptotically \ell_p FDD's

December 1st: Graham Brightwell (London School of Economics) Non-transitive sets of dice

Previous seminars

• Autumn 2004
• Spring 2004
• Autumn 2003
• Spring 2005

Abstracts

October 6th: Tony Carbery (Edinburgh) The Brascamp--Lieb inequalities: finiteness, structure and extremals

We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss. In that paper, the heat flow method was used to obtain the rank one case of Lieb's fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher rank case, giving two new proofs of the general rank case of Lieb's theorem.

October 13th: Rob Curtis (Birmingham) The Leech lattice $\Lambda$ and the Conway group $\cdot O$ revisited

The Leech lattice $\Lambda$ was discovered by John Leech in 1965 in connection with the packing of identical spheres into 24--dimensional space ${\mathbb R}^{24}$ so that their centres lie at the lattice points. Its construction exploits the rich combinatorial structure underlying the Mathieu group M_{24}. The Conway group $\cdot O$ is the group of automorphisms of the Leech lattice (fixing the origin). It was clear from the construction that $\Lambda$ was preserved by M_{24} and by sign changes on coordinate positions corresponding to words in the binary Golay code. John Conway, with computational assistance from Mike Guy, was able to produce elegant additional elements which he labelled $\xi_T$, for T a tetrad of coordinate positions, and then obtain the order of the group of automorphisms and prove it to be perfect.

In this talk we shall turn the process around and use the methods of symmetric generation of groups to obtain $\cdot O$ directly from M_{24}. We shall then obtain $\Lambda$ from $\cdot O$, thus allowing the group to do the work for us.

A brief description of the necessary properties of the Mathieu group will be given, and no knowledge of symmetric generation will be assumed.

October 20th: Christian Elsholtz (Royal Holloway) Additive decompositions of the set of primes and new methods in prime number theory

This is a survey on Ostmann's problem. Ostmann asked whether there exist two sets A and B (with at least two elements each) so that their sumset A+B equals the set of primes, for sufficiently large primes. Using a new version of the large sieve method I can show, that such sets A and B would need to have counting functions of size N^(1/2 +o(1)), whereas previously only a lower bound of N^(o(1)) and an upper bound of N^(1+o(1)) was known. This implies, for example, that the set of primes cannot be decomposed into three such sets. We also look at very thin sets of primes such as primes of the form x^2+y^4 and show that underlying additive structures exist which are larger than one might have expected.

This talk will give a nontechnical survey of the underlying ideas and show how a new type of the large sieve method and combinatorial counting arguments (including graph theory) can be applied to such problems.

November 3rd: Peter Cameron (Queen Mary University of London) Counting orbits on colourings, flows, and other things.

A wide range of combinatorial counting problems are solved by suitable specialisations of the (two-variable) Tutte polynomial: these include colourings and flows in graphs, weight enumerators of codes, Jones polynomials of knots, etc. Suppose that the data includes a group of automorphisms of the object in question, and we want to count its orbits on the appropriate configurations. Is there a similar polynomial to do the job? Work in progress (with Bill Jackson and Jason Rudd) throws some light on this, and produces polynomials (in some case with many variables) for some of these questions.

November 10th: Chris Good (Birmingham) Inhomogeneities in inverse limits of dynamical systems

Even very simple maps on very simple spaces, for example quadratic maps on the unit interval, can exhibit very complex dynamics. By passing to an inverse limit space, one translates the complex dynamics to a the action of a homeomorphism, but on a complicated space.

One way to distinguish between two such spaces is to compare their sets of inhomogeneities, i.e. the points that are not locally homeomorphic to the Cantor Set cross $(0,1)$. The set of inhomogeneities can be finite, countably infinite, a Cantor set or the whole space. A good deal of work into the finite and uncountable cases but almost nothing has been done in the countably infinite case. We prove a surprising restriction on the structure of these countable sets of inhommogeneities that turns out to characterize them. The results shed some light on Ingram's Conjecture on tent maps.

Joint work with: Brian Raines (Baylor, Texas) and Robin Knight (Oxford)

November 17th: Raphael Rouquier (Leeds) Dunkl operators and Hecke algebras 

I will introduce a deformation of the ordinary derivation of real functions of one variable and discuss the corresponding operator on polynomials and on analytic functions. Then, I will switch to the dimension n case and focus on the action on polynomial functions of n variables. This is controlled by an algebra deforming the algebra of polynomial differential operators, whose representation theory is studied in analogy with the representation theory of Lie algebras. This last part brings Knizhnik-Zamolodchikov equations, braid groups, Hecke algebras.

December 1st: Graham Brightwell (London School of Economics) Non-transitive sets of dice

It is well-known that there is a trio of "three-sided dice" A, B and C, with the following property. If A and B are rolled, then the probability that the number showing on die A is greater than that on die B is strictly greater than 1/2 -- A beats B -- while similarly B beats C and C beats A. There is also a set of seven three-sided dice such that, for any pair of them, there is a third that beats both.

However, we show that, no matter how many three-sided dice we have, there will always be three dice in the collection that cannot be simultaneously beaten -- a "dominating set" of size three. Generally, we investigate the relationship between the number of sides on the dice and the minimum size of a dominating set. The tools used are mostly combinatorial, with a little elementary game theory and some combinatorial geometry thrown in.

(Joint work with Noga Alon, Hal Kierstead, Sasha Kostochka and Peter Winkler.)

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