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.