Fall 2012 Texas Geometry and Topology Conference Abstracts

Winning sets for Schmidt games, badly approximable real numbers, rational billiards and geodesics in moduli space, Howard Masur (University of Chicago)

In the 1970's Schmidt introduced a game to be played between two players in Euclidean space. Winning sets for these games have nice properties. The main example Schmidt considered were "Diophantine" reals that are badly approximated by rationals, or equivalently, flow lines on a torus badly appproximated by closed curves. It is classical that these reals correspond to geodesics that stay in a compact set in the moduli space of lattices. They also have an interpretation in terms of orbits in a square billiard table. In joint work with Jon Chaika and Yitwah Cheung we consider general rational billiards and the corresponding translation surfaces. There is a corresponding notion of a Diophantine flow direction on the surface. It is one that is badly approximated by saddle connections. We show that the set of badly approximated flow directions form a winning set for the Schmidt game.

I will discuss all of these ideas: Schmidt's game, the basic example of Diophantine reals, the connection with billiards in a square before considering more general rational billiards.

Stable cohomology of the perfect cone toroidal compactification of A_g, Samuel Grushevsky (Stony Brook University)

The cohomology of A_g, the moduli space of principally polarized complex g-dimensional abelian varieties, is the same as the cohomology of the symplectic group with integer coefficients Sp(2g,Z). The stable cohomology H^k(A_g) for g>k was computed by Borel; the stable comology of the Satake-Baily-Borel compactification of A_g was computed by Charney and Lee using topological methods. In a joint work with Klaus Hulek and Orsola Tommasi we show that the cohomology of the perfect cone toroidal compactification of A_g stabilizes, and compute some of this stable cohomology using algebro-geometric methods. There is also an independent related work in progress, by Jeffrey Giansiracusa and Gregory Sankaran, using topological methods.

Stability in the unstable cohomology of mapping class groups, SL_n(Z), and Aut(F_n) , Tom Church (Stanford University)

For each of the sequences of groups in the title, the k-th rational cohomology is known to be independent of n in a linear range n >= c*k. Furthermore, this "stable cohomology" has been explicitly computed in each case. In contrast, very little is known about the unstable cohomology, which lies outside this range.

In this talk I will explain a conjecture on a new kind of stability in the cohomology of these groups, joint with Benson Farb and Andrew Putman. These conjectures concern the unstable cohomology, in a range near the "top dimension" (the virtual cohomological dimension), and for SL_n(Z) they imply that the unstable cohomology vanishes in that range. One key ingredient is a version of Poincare duality for these groups based on the topology of the curve complex and the algebra of modular symbols. Very recently, Avner Ash has announced a proof of our conjecture for SL_n(Z) (unpublished).

An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, Sarah Koch (Harvard University)

This is joint work with John H. Hubbard.

We outline a proof that the Deligne-Mumford compactification of the moduli space of curves is isomorphic (as an analytic space) to the quotient of augmented Teichmueller space by the action of the mapping class group. The main difficulty is putting a complex structure on this quotient.

Classification of universal Jacobians over the moduli space of curves, Gavril Farkas (Humboldt University)

The universal Jacobian J_g is the fibration over the moduli space of curves with fibres being Jacobian varieties of curves of genus g. I will discuss joint work with Verra in which a complete classification of J_g by Kodaira dimension has been carried out. Thus J_g is a unirational variety when g<9, has Kodaira dimension zero (respectively 19) when g=10 (respectively g=11), and is of Kodaira dimension 3g-3 for all other cases.

Effective Veech dichotomy and diagonal flows, Matthew Bainbridge (Indiana University)

The Veech dichotomy tells us that for certain very symmetric flat surfaces, we have a nearly perfect understanding of the dynamics of the geodesic flow: in every direction, the geodesic flow is either uniquely ergodic or periodic. In this talk, we discuss a more effective version of the Veech dichotomy which allows one to control the geometry of periodic orbits in any direction. The proof involves geometry of numbers, and dynamics of diagonal flows on homogeneous spaces over the adeles. This is joint work with Martin Möller.

Cohomological amplitude of moduli spaces of curves, Eduard Looijenga (University of Utrecht)

We show that the cohomological amplitude of the universal curve of genus g>0 is at most g-1. This implies the theorems of Harer on the homotopy of the moduli spaces of curves as well as Diaz's theorem which states that any complete subvariety of M_g (g>1) can have dimension at most g-2.