Upcoming Talks

By a log-symplectic form I mean a meromorphic closed non-degenerate 2-form omega on a complex manifold X that has only simple poles along a divisor D. I will discuss how the de Rham class of omega in H^2(X\D) determines the deformations of the triple (X,D,omega). Some open problems about topology of the divisor deforming D will be discussed.

In joint work with Christopher Frei we use the circle method to estimate averages of arithmetic functions over values of random integer polynomials.
 

Given a hyperbolic curve $Y$ defined over the integers and a finite set of primes $S$, the set of $S$-integral points $Y(\mathbb{Z}_S)$ is finite by theorems of Siegel, Mahler, and Faltings. Determining this set in practice is a difficult problem for which no general method is known. In this talk I report on joint work in progress with Martin Lüdtke in which we develop a Chabauty--Coleman method for finding $S$-integral points on affine curves.

In the past decade, a major triumph for the Tate conjecture (over finite fields) is its resolution for K3 surfaces. However, all known proofs rely crucially on the Kuga-Satake construction—effectively outsourcing certain difficulties to the theory of abelian varieties. There is an alternative approach which seeks to prove the conjecture by linking it to finiteness statements in arithmetic geometry, and using only the geometry of K3 surfaces.