(September 10) Stephen Halperin: Lusternik-Schnirelmann category - LS category is a homotopy invariant of topological spaces that was introduced because it provides a lower bound for the number of critical points of a smooth function on a manifold. More recently it has turned out to play a significant role in determining the structure of the homotopy Lie algebra of the space. This talk will give a broad survey of the properties of LS category and its applications in homotopy theory.

(September 24) Lars Wahlbin: A priori and a posteriori error estimates in finite element approximation. - Intended for a general audience (no knoweledge of the finite element method is assumed). I shall try to explain the difference between a priori and a posteriori error estimates (and indicators) and I shall selectrively survey methods and results.

(October 1) Matt Foreman: A Descriptive View of Ergodic Theory - We survey some results in Ergodic theory from the point of view of Descriptive Set Theory, illustrating with examples how the invariant of `logical complexity' can be used as a tool to destinguish between classes of measure preserving transformations.

(November 5) M.A. Kaashoek: Unbounded linear operators: an example and its applications - Unbounded linear operators have been among the main fields of interest of Professor Seymour Goldberg. His 1966 book [1], which reappeared in a Dover edition about ten years ago, still serves as a very useful and often cited source of information on this topic. The present talk concerns a class of unbounded operators related to ordinary differential operators on the half line which was introduced in [2]. The operators involved differ considerably from thier counterparts on a finite interval and do not have a compact resolvant. Their spectra and essential spectra will be described. Also, the Fredholm characteristics will be explicitly identified. As an application, explicit inversion formulas will be obtained for systems of Wiener-Hopf integral equations with a rational symbol.

[1] S. Goldberg, Unbounded Linear Operators, Theory and Applications, McGraw-Hill, New York, 1966.

[2] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, Volume 1, Birkhauser Verlag, Basel, 1990.

(November 10) Richard A. Shore: Computable Structures -- Presentations Matter - The following is a common situation in mathematics:

We are given some mathematical structure (say a vector space V), and we prove the existence of some desired element or set of elements (say a basis for V). We then have the following question -- can we obtain such a set effectively (or, computably), i.e. is there an algorithm which lists its elements (the elements in a basis for V in our example)? As a similar example, given an algebraically closed field F, can we test for algebraic independence effectively, i.e. is there an algorithm for doing this?

Certainly if we are to be able to do this we must first be given the original structure (the vector space V or field F) computably, that is there must be algorithms for computing the given domain (i.e., V or F) and the operations on it (+ and scalar multiplication for V, + and x for F). Such a structure is called computable.

In some cases the answer to the question is an unequivocal "Yes" -- for example, we can always effectively find a basis for a finite dimensional vector space (over Q), and the algorithm is given in any basic linear algebra course. In other cases the answer depends on the precise way in which the original structure is given -- this happens in the case of finding a basis for an infinite dimensional vector space. The problem is that there may be different presentations (i.e. algorithms) that describe the same structure (the vector space or field) up to isomorphism.

In this survey talk we give simple examples of such situations, with proofs, and then describe some general results about when the existence of computable solutions to certain types of problems is independent of the presentation and when not. If they are not independent of the presentation we give some indications of how badly they may vary. Finally we describe how these possibilities are realized in various classes of mathematical structures such as linear orders, abelian groups, nilpotent groups, algebraically closed fields, and real closed fields.

(November 12) Ravi Ramakrishna: Deforming global Galois representations - The title of my talk is: Deforming global Galois representations. Serre has conjectured that mod p Galois representations "come from" modular forms. A special case of a conjecture of Fontaine and Mazur can be viewed as a characteristic zero analog of Serre's conjecture. We will explain these conjectures and work relating them.

(February 4) Bill Goldman: Flat Lorentzian 3-manifolds and hyperbolic geometry - Unlike Euclidean crystallographic groups, properly discontinuous groups of affine transformations need not be amenable. For example, a free group of rank two admits a properly discontinuous affine action on 3-space. Milnor imagined how one might construct such an action: deform a Schottky subgroup of O(2,1) inside the group of Lorentzian isometries of Minkowski space, although as he wrote in 1977, ``it seems difficult to decide whether the resulting group action is properly discontinuous.'' In 1983, Margulis, while trying to prove such groups don't exist, constructed the first examples. In his 1990 doctoral thesis, Drumm constructed explicit geometric examples from fundamental polyhedra. In this talk I hope to convince you these groups do actually exist. Their rich structure is quite mysterious, and I hope to mention questions concerning their deformation theory, dynamical properties of geodesics and waves on the quotient space-times, and arithmetic questions.

(February 11) Anatole Katok: Rigidity in ergodic theory and class numbers of algebraic number fields - ***Based on joint work with Svetlana Katok and Klaus Schmidt***

The most celebrated result of classical ergodic theory, the Ornstein isomorphism theorem for Bernoulli shifts, has far-reaching implications concerning extreme softness or rahter placticity of measurable orbit structure for most ``natural'' dynamical systems. The totalities of isomorphisms, cetralizers, factors, square roots etc for such systems are huge and amorphous, and measurable orbit structure has no connections with finer structures such as topological, duifferentiable, homogeneous, etc.

The picture miraculously changes when one passes to dynamical systems with multi-dimensional time, i.e. actions of Z^k or R^k for k>1. Very simple and natural examples such as the actions of Z^k by automorphisms of a torus display remarkable rigidity of the measurable orbit structure. Under nutural irreducubility assumptions every measurable isomorphism between such actions coincides almost everywhere with an affine map; similar rigidity holds for cetralizers, factors etc. These facts allow us to bring commutative algebra and algebraic number theory methods into play.

As an illustration we will discuss the following example. Two actions of Z^k by toral automorphisms which are isomorphic over Q (or over R) are weakly isomorphic; each of them is a factor of the other with a finite fiber. This fact can be understood without any measure theory by just considering the actions on various lattices. Since elements of the actions are Bernoulii, for k=1 this implies measure-theoretic isomorphism due to the Ornstein theorem. Equivalence classes (over Z) of integer matrices isomorphic over Q are closely related to classes of fractional ideals in algebraic number fields. Using information about class numbers of totally real cubic fields one produces actions of Z² on T³ which are weakly isomorphic but not isomorphic and cannot be distinguished by natural invariants (entropy function, cetralizers etc).

(February 18) Harry Tamvakis: The Hodge star operator on Schubert forms - The Hodge star operator acts on the space of differential forms on a hermitian complex manifold X and induces an action on the harmonic forms, hence on the cohomology of X. I will give explicit combinatorial formulas for this action when X is a compact hermitian symmetric space, such as a Grassmannian, and discuss connections of this problem with the Lefschetz theory on X. I will also talk about the motivation for this work, which came from a conjecture in Arakelov geometry.

(March 10) David Marker: Nonstandard models and asymptotic behaviour - I will describe the construction of a nonstandard model of the real field with exponentiation and show how to use it to solve a problem of Hardy.

(March 15) Vidar Thomée: Stepping in parabolic problems - Approximation of analytic semigroups - We consider the discretization in time of the equation u'+Au=0 where A is a positive definite operator in a Hilbert space. The time discretization is based on applying the operator r(kA) at each time step where r(z) is a rational function and k the time increment. For such schemes we show stability and error bounds under various assumptions. The discussion is generalized to the case that -A is a generator of an analytic semigroup in a Banach space. Applications are given to fully discrete methods for parabolic partial differential equations where finite element approximations are used in the spatial variable.

(April 7) A. Tsavaras: Hydrodynamic limits and the kinetic formulation of systems of Conservation Laws - This talk will present a survey of recent results on the subject of hydrodynamic limits for continuous and discrete kinetic models, in situations where the limit is a scalar or a system of two hyperbolic conservation laws. We will discuss the natural structure of these problems through variants of the Boltzmann H-theorem, the existence (in certain cases) of stronger dissipative structures and the connection with the kinetic formulation for conservation laws.

(April 28) M. Ram Murty: A Century of Zeta Functions - We will discuss how the notion of a zeta function has revolutionised number theory in the past century. We will emphasize how the analytic theory and algebraic theory coalesce in the theory of the zeta function. This talk will be accessible to undergraduates.

(May 5) Anand Pillay: Geometric model theory and applications in number theory and geometry - Classification Theory and subsequently geometric model theory have developed a substantial amount of machinery, tools and concepts for analysing first order structures. Many of the notions turn out to have meaning and yield results in areas such as number theory (rational points), the algebraic theory of differential equations, and the classification of compact complex manifolds. I will discuss some of these developments.