Matt Kerr - Differential equations, hypergeometric families, and algebraic cycles - 12.06.2023
Channel: Hörsaal Mathematikon
Date: 2023-06-13

Hodge theory, Mirror Symmetry, and Physics of Calabi-Yau Moduli
https://web.mathi.uni-heidelberg.de/cymod23
A MATCH-PIMS-STRUCTURES workshop in Mathematical Physics, held from June 12 to 16, 2023, at the Mathematikon in Heidelberg
For more talks visit https://www.youtube.com/playlist?list=PL57PsbjMibmQ37JSKzyxqvmIhyAfF1iFs

Abstract: We discuss two approaches to solving inhomogeneous equations of the form L(.)=t^(1/d), where L is a hypergeometric differential operator attached to a family of CY varieties. The first is by elementary complex analysis, using so-called Frobenius deformations, and gives an explicit series solution.
The second is via normal functions attached to algebraic cycles (both "classical" and "higher") on a base-change of the family. I will briefly review regulator maps, their relation to inhomogeneous Picard-Fuchs equations, and the relevant cases of the Beilinson conjecture. Turning then to the CY3 examples classified by Doran-Morgan, I will explain how to identify which types of cycles arise (viz., K_0​, K_2​, or K_4​ classes), and how to identify the cycles in certain cases.
This is the first half of a joint talk with V. Golyshev, whose part will discuss the Beilinson conjecture aspect in greater depth.