Speaker
Description
Algebraic cycles on Calabi-Yau 3-folds played a starring role in demonstrating the inequivalence of algebraic and homological equivalence, opening mirror symmetry on the quintic, and describing asymptotics of local Gromov-Witten invariants. In this talk, they play the role of a new testing ground for connections between transcendental and arithmetic geometry.
On the transcendental side, we have the extensions of mixed Hodge structure that arise from the motivic cohomology / higher Chow / algebraic K-theory groups of a variety over Q. Their periods define the regulators and height pairings which, for a family of varieties, produce solutions of inhomogeneous Picard-Fuchs equations. Thinking in terms of families deforming the original variety turns out to be the key to going beyond “classical” cases, both for constructing such cycles and computing their transcendental invariants.
The arithmetic of a variety over Q is described in part by the L-function attached to the Galois representation on its middle cohomology. The Bloch-Beilinson conjectures, which posit a relationship between its special values and the generalized periods above, remain one of the deepest open problems in mathematics. In this talk, based on recent work with Vasily Golyshev, I’ll explain how to use hypergeometric families and Hadamard convolutions to produce new numerical evidence for these conjectures.
