Hilbert's 11th problem asks about the representation of integers in a number field by an integral quadratic form. We explain the problem, some of the key developments, and in particular the recent resolution of the problem.

Interfaces or phase transitions appear very often in physics, chemistry, biology, etc., for instance in interactions fluid-gas or solid-fluid, or in flame propagation in reactors. Due to surface tension, some interfaces tend to minimize its area. This is an interesting connection with the classical and rich theory of minimal surfaces. At the same time, the local study of some interfaces through a blow-up analysis leads to nonlinear elliptic equations in the whole space $\mathbb R^n$, and to rich problems on the symmetry properties of their solutions. We will discuss one of such problems; it was posed by E. DeGiorgi in 1978 and it is still widely open.

A number of results have been obtained recently which analyse greedy algorithms on regular graphs and how they perform on average. I will discuss the current use of the differential equation method for these and related problems. The examples considered include the greedy construction of small dominating sets, large induced matchings and large 2-independent sets in regular graphs. The primary aim of these algorithms has been to obtain bounds on the sizes of these sets in almost all regular graphs. Some of the results obtained recently have applications in computer science.

This talk is aimed at a general audience: specialized knowledge in graph theory, probability, differential equations or computer science is not required.

We will discuss recent advances in the mathematical understanding of critical percolation and related models, which allowed to prove some of the predictions originating in physics.

Quantum computers are hypothetical devices which use the principles of quantum mechanics to perform computations. For some difficult computational problems, including the cryptographically important problems of prime factorization and finding discrete logarithms, the best algorithms known for classical computers are exponentially slower than the algorithms known for quantum computers. Although they ave not yet been built, quantum computers do not appear to violate any fundamental properties of physics. I give a mathematical model of quantum computation, explain how quantum mechanics provides this extra computational power, and briefly describe the algorithm for efficient prime factorization.

Because wavelets are unconditional bases for many function spaces, they give rise to good nonlinear approximation bounds. These have been viewed by mathematicians as the "explanation" of why wavelets work well for compression in a variety of settings. Yet the practice of compression is different from the question of how well we can approximate in a given basis if we are allowed to keep any N terms we choose -- in practice, one has to measure the (expected) distortion as a function of the total number of bits used to convey the information. It turns out that in order to exploit the nonlinear approximation results, one has to combine them with "smart" coding strategies (invented already by the engineers!). On the other hand, the concrete algorithms suggest yet different spaces for approximation. This is joint work with Albert Cohen, Wolfgang Dahmen, Ron DeVore, Onur Guleryuz, and Michael Orchard.

What return should you expect when you take on a given amount of risk? How should that return depend upon other people's behavior? What principles can you use to answer these questions? We discuss approaches to these issues in the field of financial modeling.

Schedule for Previous years.

Last modified: April 16, 2002