Schedule of Talks

! All talks will take place in the Mathematics Department (McAllister Building), room 114 !

K. Burns: The talk will descibe interactions between two approaches to the study of the geodesics joining two points in a Riemannian manifold.  One of them relates the growth of the number of these geodesics as their length increases to the topological entropy of the geodesic flow. The other studies whether or not these geodesics can be blocked by a finite number of point obstacles.

K. Kuperberg:  A trajectory of a flow on a 3-manifold is wild if the closure of at least one of the semi-trajectories is a wild arc. A trajectory is 2-wild if the closure of each semi-trajectory is a  wild arc. A flow is a continuous R-action on manifolds. We show that:
1. Every closed and connected 3-manifold admits a flow with exactly one fixed point and whose every non-trivial trajectory is 2-wild. 
2. Every boundaryless 3-manifold admits a flow with a discrete set of  fixed points and every non-trivial trajectory 2-wild.

E. Lindenstrauss: Let Γ be a Zariski dense subgroup of SL(n,Z), S a finite set of generators for Γ, and µ a probability measure on S. We study the µ-stationary as well as the Γ-invariant measures by combining methods of additive combinatorics, particularly Bourgain's proof of the discretized ring conjecture, with simple facts regarding the action of Γ on the boundary. In particular we prove a conjecture of Furstenberg regarding the stiffness of such actions. (Joint work with Bourgain, Furman, Mozes.)

J. Millson: I will report on joint work with Ben Howard and Chris Manon concerning toric  varieties associated to the space of polygons in Euclidean three-space with fixed side-lengths modulo orientation-preserving congruence.

In 1996 Misha Kapovich and I (and Klyachko independently) proved (JDG Vol. 44) that in case the side-lengths  are integers these spaces of polygons are complex projective varieties  of complex dimension n-3.  We introduced Hamiltonian bending flows associated to diagonals of the polygon .For each choice of  n-3 nonintersecting diagonals we obtained n-3 commuting periodic Hamiltonian flows so these moduli spaces are almost toric (except the flows are not  everywhere defined). Two Japanese mathematicians Y. Kamiyama and T. Yoshida noted that by coarsening the  notion of congruence of polygons one can make the bending flows everywhere defined but the resulting collapsed polygon spaces are no longer manifolds.

Now  a ring-theoretic construction due to Sturmfels  can be used to construct toric degenerations of the above moduli spaces of polygons for the case of integer side-lengths. The point of my talk will be to prove that the (special fibers of the) toric degenerations of Sturmfels coincide as topological spaces (with compact torus action) with the collapsed polygon spaces of Kamiyama and Yoshida - so these latter spaces are in fact toric varieties.

P. Sarnak: We review various classical problems which are concerned with looking for primes or numbers with few prime factors and then put these in a geometric and group theoretic context of values at polynomials on orbits of an action of affine space which preserves Z^n. The development of the combinatorial sieve in this context presents a number of novel features one of which is that certain graphs associated with these orbits be "expanders". We will give applications to classical problems such as the divisibilty of the areas of pythagorean triangles.

R. Schwartz: Outer billiards is a simple dynamical system in the plane based on a convex shape. B.H. Neumann introduced outer billiards in the 1950s and J. Moser popularized it in the 1970s. All along, one of the central questions has been: Does there exist a convex shape for which outer billiards has an unbounded orbit? Recently I proved that outer billiards has an unbounded orbit when defined relative to the Penrose kite, the symmetric convex quadrilateral that appears in the Penrose tiling. I will touch upon this result in my talk, but mainly I will describe a connection I recently discovered between outer billiards on kites and the modular group. One corollary of this connection is that there are uncountably many different kites relative to which outer billiards has unbounded orbits.

H. Stark: This talk represents joint work with Andrew Granville. In a 2000 paper, Granville and I showed that the best possible formulation of a uniform ABC conjecture over number fields applied to the modular function j(z) at CM points delivers a result provable by the generalized Riemann hypothesis. In this talk we investigate the same situation, but with the classical lambda function in place of j(z). The interest in this case is that the potential is there to come in with an ABC result which  is too strong. Such an outcome would cast doubt on the uniform ABC conjecture. In fact the ultimate result is again precisely at the Riemann hypothesis prediction, but it is quite interesting how this happens. The lambda result also disproves an alternative version of the uniform ABC Conjecture.

A. Terras:  This talk will survey joint work with Harold Stark on zeta and L-functions of connected graphs and their coverings. These zeta functions are simultaneously analogs of Dedekind zeta functions of number  fields and Selberg zeta functions of a Riemannian manifold. They also are special cases of the Ruelle zeta function of a dynamical system. They are reciprocals of polynomials and thus have only poles, no zeros. In particular, I will consider the distribution of poles.