Past

My lecture is based on results of [1] and [2]. In [1] we use an extension of the method due to Borel and Prasad to determine the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group. In [2] the results of [1] are combined with the previously known asymptotic of the number of subgroups in a given lattice in order to study the general lattice growth. We show that for many high-rank simple Lie groups (and conjecturally for all) the rate of growth of lattices of covolume at most $x$ is like $x^{\log x}$ and not $x^{\log x/ \log\log x}$ as it was conjectured before. We also prove that the

conjecture is still true (again for "most" groups) if one restricts to counting non-uniform lattices. A crucial ingredient of the argument in [2] is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.

I plan to give an overview of these recent results and discuss some ideas beyond the proofs.

[1] M. Belolipetsky (with an appendix by J. Ellenberg and A.

Venkatesh), Counting maximal arithmetic subgroups, arXiv:

math.GR/0501198.

[2] M. Belolipetsky, A. Lubotzky, Class field towers and subgroup

growth, work in progress.

Two dimensional minimal cones were fully classified by Jean Taylor in the mid

70's.  In joint work with G. David and T. De Pauw we prove that a closed

set which is close to a minimal cone at all scales and at all locations is

locally a bi-Hoelder image of a minimal cone.  This result is analogous to

Reifenberg's disk theorem.  A couple of applications will be discussed.

This talk will be a tour of a couple of problems in the differential topology of

loop spaces.  We shall do a "compare and contrast" between these problems

and their finite dimensional analogues, with the aim of illustrating some of the

intriguing aspects of infinite dimensional manifolds.

The problems that we shall focus on are those of defining analogues of

differential operators on manifolds, in particular the Dirac and the

(semi-infinite) de Rham operators.

First of all, I intend to remind us of several properties of

polyhedral cones and cone-generated orders which will be used for constructing Pareto sets in multiple objective optimisation problems.

Afterwards, I will consider multiple objective discounted Markov Decision Process. Methods of Convex Analysis and the Dynamic Programming Approach allow one to construct the Pareto sets and study their properties. For instance, I will show that in the unichain case, Pareto sets for different initial distributions are topologically equivalent. Finally, I will present an example on the optimal management of a deteriorating system.

We examine the classical orthogonal polynomial ensembles using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. Equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Applications to sharp deviation inequalities on largest eigenvalues are discussed.

Direct methods for solving large sparse linear systems of equations are popular because of their generality and robustness. As the requirements of computational scientists for more accurate models increases, so inevitably do the sizes of the systems that must be solved and the amount of memory needed by direct solvers.

For many users, the option of using a computer with a sufficiently large memory is either not available or is too expensive. Using a preconditioned iterative solver may be possible but for the "tough" systems that arise from many practical applications, the difficulties involved in finding and computing a good preconditioner can make iterative methods infeasible. An alternative is to use a direct solver that is able to hold its data structures on disk, that is, an out-of-core solver.

In this talk, we will explain the multifrontal algorithm and discuss the design and development of a new HSL sparse symmetric out-of-core solver that uses it. Both the system matrix A and its factors are stored externally. For the indefinite case, numerical pivoting using 1x1 and 2x2 pivots is incorporated. To minimise storage for the system data, a reverse communication interface is used. Input of A is either by rows or by elements.

An important feature of the package is that all input and output to disk is performed through a set of Fortran subroutines that manage a virtual memory system so that actual i/o occurs only when really necessary. Also important is to design in-core data structures that avoid expensive searches. All these aspects will be discussed.

At the time of writing, we are tuning the code for the positive-definite case and have performance figures for real problems. By the time of the seminar, we hope to have developed the indefinite case, too.

In this talk we will discuss a theory of divergence-measure fields and related

geometric measures, developed recently, and its applications to some fundamental

issues in mathematical continuum physics and nonlinear conservation laws whose

solutions have very weak regularity, including hyperbolic conservation laws,

degenerate parabolic equations, degenerate elliptic equations, among others.

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence.

Moving hydrodynamic boundaries (waves and bubbles, for example) produce acoustic signatures.

It is well known that the computation of a few extreme eigenvalues, and the corresponding eigenvectors, of a symmetric matrix A can be rewritten as computing extrema of the Rayleigh quotient of A. However, since the Rayleigh quotient is a homogeneous function of degree zero, its extremizers are not isolated. This difficulty can be remedied by restricting the search space to a well-chosen manifold, which brings the extreme eigenvalue problem into the realm of optimization on manifolds. In this presentation, I will show how a recently-proposed generalization of trust-region methods to Riemannian manifolds applies to this problem, and how the resulting algorithms compare with existing ones.

I will also show how the Joint Diagonalization problem (that is, approximately diagonalizing a collection of symmetric matrices via a congruence transformation) can be tackled by a differential geometric approach. This problem has an important application in Independent Component Analysis.

For atomistic (and related) material models, global minimization

gives the wrong qualitative behaviour; a theory of equilibrium

solutions needs to be defined in different terms. In this talk, a

process based on gradient flow evolutions is presented, to describe

local minimization for simple atomistic models based on the Lennard-

Jones potential. As an application, it is shown that an atomistic

gradient flow evolution converges to a gradient flow of a continuum

energy, as the spacing between the atoms tends to zero. In addition,

the convergence of the resulting equilibria is investigated, in the

case of both elastic deformation and fracture.

Levy trees are random continuous trees that are obtained as

scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in

PTRF) math.PR/0509518 (preprint)

math.PR/0509690 (preprint).

I will present two-sided estimates for the heat kernel of the elliptic

Schr