It is believed that there are universal groups that govern fundamental processes in number theory, automorphic forms and algebraic geometry. Some such groups are well known, while others are only hypothetical. We shall use these groups as a means of discussing some of the basic questions in the theory of automorphic forms.

The complexity of the instruments traded in the energy markets, together with the extreme volatility of electricity prices offer challenging problems to the mathematicians. We shall review some of the practical issues with the gas and power markets (spark spread options and plant valuation, gas storage, weather derivatives, swing options), and we shall formulate the corresponding mathematical problems. After discussing the mathematics involved in the existing solutions, we will concentrate on options with multiple American exercises, and we will present new mathematical results for the pricing of these options.

Surgery operations like ``connected sum'' have been standard tools in differential geometry for many years. In the last decade two operations of this type: ``cutting'' and ``gluing'' have become standard tools in symplectic geometry as well. In this lecture I will describe a number of recent developments in which these operations have played a role, among them the solution of the ``quantization commutes with reduction'' conjecture, an elementary proof of the Kirwan convexity theorem and the construction of many interesting examples of non-Kaehlerizable symplectic manifolds.

In mathematical work on quantum mechanics we are often interested in the density of states in the semi-classical limit. The work of Sjöstrand on modified Weyl upper bounds showed a relation between the density of states in quantum chaotic scattering and the dimension of the classical trapped set.

This work motivated recent rigorous and numerical work on quantum resonances in chaotic scattering, in particular estimates on classical dynamical zeta functions for Schottky groups, where the trapped set is related to the limit set of the group.

In my talk I will explain these concepts and present the recent numerical results in potential, obstacle, and geometric scattering (joint work with L. Guillopé, K. Lin, W. Lu, and S. Sridhar).