Past

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Alan Turing introduced the sensitivity of the solution of a numerical problem to changes in its data as a way to measure the difficulty of solving the problem accurately. Condition numbers are now considered fundamental to sensitivity analysis. They have been used to measure the mathematical difficulty of many linear algebra problems, including linear systems, linear least-squares, and eigenvalue problems. By definition, unless exact arithmetic is used, it is expected to be difficultto accurately solve an ill-conditioned problem.

In this talk we focus on least-squares problems. After a historical overview of condition number for least-squares, we introduce two related condition numbers. The first is the partial condition number, which measures the sensitivity of a linear combination of the components of the solution. The second is related the truncated SVD solution of the problem, which is often used when the matrix is nearly rank deficient.

Throughout the talk we are interested in three types of results :closed formulas for condition numbers, sharp bounds and statistical estimates.

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.