Georgia Institute of Technology College of Sciences

Billiards is a dynamical system generated by an uniform motion of a point particle (ray of light, sound, etc.) in a domain with piecewise smooth boundary. Upon reaching the boundary the particle reflected according to the law "the angle of incidence equals the angle of reflection". Billiards appear as natural models in various branches of physics. More recently this type of models were used in oceanography, operations research, computer science, etc. I'll explain on very simple examples what is a regular and what is chaotic dynamics, the mechanisms of chaos and natural measures of complexity in dynamical systems. The talk will be accessible to undergraduates.

Let k be a global field or a local field. Class field theory says that every central division algebra over k is cyclic. Let l be a prime not equal to the characteristic of k. If k contains a primitive l-th root of unity, then this leads to the fact that every element in H^2(k, µ_l ) is a symbol. A natural question is a higher dimensional analogue of this result: Let F be a function field in one variable over k which contains a primitive l-th root of unity. Is every element in H^3(F, µ_l ) a symbol? In this talk we answer this question in affirmative for k a p-adic field or a global field of positive characteristic. The main tool is a certain local global principle for elements of H^3(F, µ_l ) in terms of symbols in H^2(F µ_l ). We also show that this local-global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields.

A useful parametrization of the one variable trigonometric moment problem is given in terms of polynomials orthogonal on the unit circle. A description of this parameterization will be given as well as some of its uses. We will then describe a possible two variable extension.

In 1854 Riemann extended Gauss' ideas on curved geometries from two dimensional surfaces to higher dimensions. Since that time mathematicians have tried to understand the structure of geometric spaces based on their curvature properties. It turns out that basic questions remain unanswered in this direction. In this lecture we will give a history of such questions for spaces with positive curvature, and describe the progress that has been made as well as some outstanding conjectures which remain to be settled.