The geometry and topology of rational surfaces with an anticanonical cycle

Robert Friedman
Columbia University
November 18, 2014
Let \(Y\) be a smooth rational surface and let \(D\) be an effective divisor linearly equivalent to \(-K_Y\), such that \(D\) is a cycle of smooth rational curves. Such pairs \((Y,D)\) arise in many contexts, for example in the study of degenerations of \(K3\) surfaces or in the theory of deformations of minimally elliptic singularities. Deformation types of such pairs come with two extra pieces of structure: the “generic” ample cone, i.e.

Birational geometry of complex hyperbolic manifolds

Gabriele di Cerbo
Columbia University
November 19, 2014
In 1984 Hirzebruch constructed the first examples of smooth toroidal compactifications of ball quotients with non-nef canonical divisor. In this talk, I will show that if the dimension is greater or equal than three then such examples cannot exist. We will use this result to reprove and improve classical theorems, such as boundedness of hyperbolic manifolds, Baily-Borel embeddings and cusps count.

Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets

Matthew Young
Texas A & M University; von Neumann Fellow, School of Mathematics
November 20, 2014
One of the major themes of the analytic theory of automorphic forms is the connection between equidistribution and subconvexity. An early example of this is the famous result of Duke showing the equidistribution of Heegner points on the modular surface, a problem that boils down to the subconvexity problem for the quadratic twists of Hecke-Maass L-functions. It is interesting to understand if the Heegner points also equidistribute on finer scales, a question that leads one to seek strong bounds on a large collection of central values.

Cyclic homology and \(S^1\)-equivariant symplectic cohomology

Sheel Ganatra
Stanford University
November 21, 2014
In this talk, we study two natural circle actions in Floer theory, one on symplectic cohomology and one on the Hochschild homology of the Fukaya category. We show that the geometric open-closed string map between these two complexes is \(S^1\)-equivariant, at a suitable chain level. In particular, there are induced maps between equivariant homology theories, natural with respect to Gysin sequences, which are isomorphisms whenever the non-equivariant map is.