Past Colloquia

4 May 2007
16:30
Professor Ben Green
Abstract
I shall report on a programme of research which is joint with Terence Tao. Our goal is to count the number of solutions to a system of linear equations, in which all variables are prime, in as much generality as possible. One success of the programme so far has been an asymptotic for the number of four-term arithmetic progressions p_1 < p_2 < p_3 < p_4 <= N of primes, defined by the pair of linear equations p_1 + p_3 = 2p_2, p_2 + p_4 = 2p_3. The talk will be accessible to a general audience.
2 March 2007
16:30
Prof. Angus MacIntyre
Abstract
&nbsp; <font SIZE="2"> Model theory typically looks at classical mathematical structures in novel ways. The guiding principle is to understand what relations are definable, and there are usually related questions of effectivity. In the case of Lie theory, there are two current lines of research, both of which I will describe, but with more emphasis on the first. The most advanced work concerns exponentials and logarithms, in both real and complex situations. To understand the definable relations, and to show various natural problems are decidable, one uses a mixture of analytic geometry with number-theoretic conjectures related to Schanuel's Conjecture. More recent work, not yet closely connected to the preceding, concerns the limit behaviour (model-theoretically), of finite -dimensional modules over semisimple Lie algebras, and here again, for decidability, one seems obliged to consider number-theoretic decision problems, around Siegel's Theorem.
3 November 2006
16:30
Professor John Neu
Abstract
Aggregation refers to the thermodynamically favoured coalescence of individual molecular units (monomers) into dense clusters. The formation of liquid drops in oversaturated vapour, or the precipitation of solids from liquid solutions, are 'everyday' examples. A more exotic example, the crystallization of hydrophobic proteins in lipid bilayers, comes from current biophysics. This talk begins with the basic physics of the simplest classical model, in which clusters grow by absorbing or expelling monomers, and the free monomers are transported by diffusion. Next, comes the description of three successive 'eras' of the aggregation process: NUCLEATION is the initial creation of clusters whose sizes are sufficiently large that they most likely continue to grow, instead of dissolving back into monomers. The essential physical idea is growth by unlikely fluctuations past a high free energy barrier. The GROWTH of the clusters after nucleation depletes the initial oversaturation of monomer. The free energy barrier against nucleation increases, effectively shutting off any further nucleation. Finally, the oversaturation is so depleted, that the largest clusters grow only by dissolution of the smallest. This final era is called COARSENING. The initial rate of nucleation and the evolution of the cluster size distribution during coarsening are the subjects of classical, well known models. The 'new meat' of this talk is a 'global' model of aggregation that quantitates the nucleation era, and provides an effective initial condition for the evolution of the cluster size distribution during growth and coarsening. One by-product is the determination of explicit scales of time and cluster size for all three eras. In particular, if G_* is the initial free energy barrier against nucleation, then the characteristic time of the nucleation era is proportional to exp(2G_*/5k_bT), and the characteristic number of monomers in a cluster during the nucleation era is exp(3G_*/5k_bT). Finally, the 'global' model of aggregation informs the selection of the self similar cluster size distribution that characterizes 'mature' coarsening.
24 February 2006
16:30
Professor Jean-Marc Gambaudo
Abstract
In the year 1858, Herman Ludwig Ferdinand von Helmholtz published in <em>Crelle's Journal</em> a deep and pioneering paper on vortex motions where the topological properties of vortex lines in a fluid motion were emphasised. This work has been a strong source of inspiration for P G Tait who settled down the foundation of knot theory and for H Poincare, the father of geometric theory of dynamical systems. As a matter of fact, by the end of the 19<sup>th</sup> century, three topics, <i>knots, flows and fluids</i> were closely related. In the last decades, the topic has been boosted by a series of new appealing problems and interesting results gathered under the name <i>Topological Methods in Hydrodynamics</i>. Our talk will start with a short trip around the pioneering works. Then we will focus on two essential recent topics:
21 October 2005
16:30
Abstract
The recent developments of Mathematical Physics have brought very new ideas about the nature of space . I will argue that we have to mix the methods of noncommutative geometry of Alain Connes with the prophetic views of Grothendieck about the so-called motives and their motivic Galois group .<br> The dream of a &quot;cosmic Galois group&quot; may soon become an established reality .<br> &nbsp;
10 June 2005
16:30
Abstract
I will give an expository account of Mathai, Melrose, and Singer [math.DG/0402329], explaining how to define the projective Dirac &quot;operator&quot; when the underlying manifold is neither spin nor spin_C, and how to define its analytic index which need not be an integer. Nevertheless, the usual index formulas apply. <i>Professor Singer will be admitted as honorary member of the London Mathematical Society just before his talk.</i>

Pages