William A. Veech is pursuing the question of existence of a prime geodesic theorem for a generic meromorphic finite norm quadratic differential on a generic closed Riemann surface. Using his recently developed theory of Siegel measures, he is led to the question of whether a related Teichmuller map action of SL(2,R) on a topological component of a stratum in the moduli space of norm one quadratic differentials does not almost have invariant vectors, in the orthocomplement of the constants relative to the Liouville measure. Veech is studying moduli spaces of flat metrics with cone singularities on punctured surfaces. For fixed cone angles, there is an identification between the Teichmuller moduli space and the moduli space of metrics up to scale which gives rise to interesting integrals over the Teichmuller moduli space. Veech is attempting to prove that these integrals, which represent natural volumes of the moduli space, are finite, using his recently developed "Delaunay partition coordinates" for moduli space. He is pursuing the question of whether every geodesic relative to the natural flat (cone) metric on the truncated icosahedron is either closed or uniformly distributed, reducing this to a question of whether a certain discrete subgroup of SL(2,R) which is associated to the truncated icosahedron is a lattice.
William A. Veech is studying periodic trajectories for a class of dynamical systems. These include systems whose descriptions require little more than high school geometry but whose analysis requires rather deep notions from disparate fields, including complex analysis, ergodic theory and representation theory of Lie groups. Location and enumeration of periodic trajectories are central problems in the theory of dynamical systems. The former may be interpreted as the problem of predicting periodicity from knowledge of intitial conditions while the latter is the problem of obtaining quantitative information about the set of all periodic trajectories, including an asymptotic formula for the number of trajectories whose period is less than T for large T. An elementary example of a system for which both questions are for the most part unresolved is the uniform motion of a pointlike particle which is contained in a planar polygonal area and which rebounds from the sides of the container according to Snell's law, i.e. "angle of incidence equals angle of rebound". To indicate the complexity of the situation it may be mentioned the the problems of location and enumeration of periodic trajectories for such motion in a regular polygon (other than equilateral triangle, square and regular hexagon) were open until only recently when they were solved by Veech.
