Past Colloquia

I shall give a gentle introduction to the cohomology of finite groups from the point of view of algebra, topology, group actions and number theory

Coefficients of the characteristic polynomial are generators of the ring of polynomial functions on the space of matrices which are invariant under the conjugation. This was generalized by Chevalley to general reductive groups. By looking closely on the centralisers, one is lead to a very natural 2-category attached to Chevalley characteristic morphism. This abstract, but yet elementary, construction helps one to understand the symmetries of the fibres of the Hitchin fibration, as well as those of affine Springer fibers.

We will also explain how these groups of symmetries are related to the notion of endoscopic groups, which was introduced by Langlands in his stabilisation of the trace formula. We will also briefly explain how the symmetry groups help one to acquire a rather good understanding of the cohomology of the Hitchin fibration and eventually the proof of the fundamental lemma in Langlands' program.

In classical economic theory, one discounts future gains or losses at a constant rate: one pound in t years is worth exp(-rt) pounds today. There are now very good reasons to consider non-constant discount rates. This gives rise to a problem of time-inconsistency: a policy which is optimal today will no longer be optimal tomorrow. The concept of optimality then no longer is useful. We introduce instead a concept of equilibrium solution, and characterize it by a non-local variant of the Hamilton-Jacobi equation. We then solve the classical Ramsey model of endogenous growth in this framework, using the central manifold theorem

A random environment (in Z^d) is a collection of (random) transition probabilities, indexed by sites. Perform now a random walk using these transitions. This model is easy to describe, yet presents significant challenges to analysis. In particular, even elementary questions concerning long term behavior, such as the existence of a law of large numbers, are open. I will review in this talk the model, its history, and recent advance, focusing on examples of unexpected behavior.

Let E be an elliptic curve defined by a cubic equation with rational coefficients.
The Sato-Tate Conjecture is a statistical assertion about the variation of the number of points of E over finite fields. I review some of the main steps in my proof of this conjecture with Clozel, Shepherd-Barron, and Taylor, in the case when E has non-integral j-invariant. Emphasis will be placed on the steps involving moduli spaces of certain Calabi-Yau hypersurfaces with level structure.

If one admits a version of the stable trace formula that should soon be available, the same techniques imply that, when E and E' are two elliptic curves that are not isogenous, then the numbers of their points over finite fields are statistically independent. For reasons that have everything to do with the current limits to our understanding of the Langlands program, the analogous conjectures for three or more non-isogenous elliptic curves are entirely out of reach.

New techniques in cell and molecular biology have produced huge advances in our understanding of signal transduction and cellular response in many systems, and this has led to better cell-level models for problems ranging from biofilm formation to embryonic development. However, many problems involve very large numbers of cells, and detailed cell-based descriptions are computationally prohibitive at present. Thus rational techniques for incorporating cell-level knowledge into macroscopic equations are needed for these problems. In this talk we discuss several examples that arise in the context of cell motility and pattern formation. We will discuss systems in which the micro-to-macro transition can be made more or less completely, and also describe other systems that will require new insights and techniques.

A lattice in the plane is a discrete subgroup in R^2 isomorphic to Z^2 ; it is unimodular if the area of the quotient is 1. The space of unimodular lattices is a venerable object in mathematics related to topology, dynamics and number theory. In this talk, I'd like to present a guided tour of this space, focusing on its topological aspect. I will describe in particular the periodic orbits of the modular flow, giving rise to beautiful "modular knots". I will show some animations

We shall begin with simple Weyl type asymptotic formulae for the spectrum of Dirichlet Laplacians and eventually prove a new result which I have recently obtained, jointly with J. Dolbeault and M. Loss. Following Eden and Foias, we derive a matrix version of a generalised Sobolev inequality in one dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schrödinger operators.

Bio: Ari Laptev received his PhD in Mathematics from Leningrad University (LU) in 1978, under the supervision of Michael Solomyak. He is well known for his contributions to the Spectral Theory of Differential Operators. Between 1972 - 77 and 1977- 82 he was employed as a junior researcher and as Assistant Professor at the Mathematics & Mechanics Department of LU. In 1981- 82 he held a post-doc position at the University of Stockholm and in 1982 he lost his position at LU due to his marriage to a British subject. Up until his emigration to England in 1987 he was working as a builder, constructing houses in small villages in the Novgorod district of Russia. In 1987 he was employed in Sweden, first as a lecturer at Linköping University and then from 1992 at the Royal Institute of Technology (KTH). In 1999 he became a professor at KTH and also Vice Chairman of its Mathematics Department. In 1992 he was granted Swedish citizenship. Ari Laptev was the President of the Swedish Mathematical Society from 2001 to 2003 and the President of the Organizing Committee of the Fourth European Congress of Mathematics in Stockholm in 2004. From January 2007 he has been employed by Imperial College London. Ari Laptev has supervised twelve PhD students. From January 2007 until the end of 2010 he is President of the European Mathematical Society.

Random planar curves arise in a natural way in statistical mechanics, for example as the boundaries of clusters in critical percolation or the Ising model. There has been a great deal of mathematical activity in recent years in understanding the measure on these curves in the scaling limit, under the name of Schramm-Loewner Evolution (SLE) and its extensions. On the other hand, the scaling limit of these lattice models is also believed to be described, in a certain sense, by conformal field theory (CFT). In this talk, after an introduction to these two sets of ideas, I will give a theoretical physicist's viewpoint on possible direct connections between them.

John Cardy studied Mathematics at Cambridge. After some time at CERN, Geneva he joined the physics faculty at Santa Barbara. He moved to Oxford in 1993 where he is a Senior Research Fellow at All Souls College and a Professor of Physics. From 2002-2003 and 2004-2005 he was a member of the IAS, Princeton. Among other work on the applications of quantum field theory, in the 1980s he helped develop the methods of conformal field theory. Professor Cardy is a Fellow of the Royal Society, a recipient of the 2000 Paul Dirac Medal and Prize of the Institute of Physics, and of the 2004 Lars Onsager Prize of the American Physical Society "for his profound and original applications of conformal invariance to the bulk and boundary properties of two-dimensional statistical systems."

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.

  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.

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.

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In the year 1858, Herman Ludwig Ferdinand von Helmholtz published in Crelle's Journal 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^th century, three topics, knots, flows and fluids 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 Topological Methods in Hydrodynamics. Our talk will start with a short trip around the pioneering works. Then we will focus on two essential recent topics:

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 .
The dream of a "cosmic Galois group" may soon become an established reality .
 

I will give an expository account of Mathai, Melrose, and Singer [math.DG/0402329], explaining how to define the projective Dirac "operator" 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. Professor Singer will be admitted as honorary member of the London Mathematical Society just before his talk.

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