RIT on Hodge Theory Archives for Fall 2024 to Spring 2025
How Markman Saves the Hodge conjecture (for Weil type abelian fourfolds) from Kontsevich
When: Fri, March 1, 2024 - 4:15pmWhere: Kirwan Hall 1311
Speaker: Patrick Brosnan (UMD) - https://www.math.umd.edu/~pbrosnan/
Abstract: I'll explain two opposing pieces of work: (1) Markman's proof of the
Hodge conjecture for general Weil type abelian fourfolds of discriminant 1, and
(2) Kontsevich's tropical approach to finding a counterexample to the Hodge
conjecture for Weil type abelian varieties. Then I'll explain why Markman's
proof of the Hodge conjecture in the discriminant 1 case rules out Kontsevich's
approach in dimension 4 (for arbitrary discriminant). This last observation is pretty elementary, but I think it illustrates some of the techniques that go into working with Mumford-Tate groups in a nice way. The observation itself is part of joint work in progress that I'm doing with Helge Ruddat.
How Markman Saves the Hodge conjecture (for Weil type abelian fourfolds) from Kontsevich 2
When: Fri, March 8, 2024 - 4:15pmWhere: Kirwan Hall 1311
Speaker: Patrick Brosnan (UMD) - https://www.math.umd.edu/~pbrosnan/
Abstract: I'll continue my talk from last week.
The Riemann-Hilbert Problem
When: Fri, March 15, 2024 - 4:15pmWhere: Kirwan Hall 1311
Speaker: Emerson Hemley (University of Maryland) -
Abstract: Given a system of linear differential equations on a space X, one gets a representation of the fundamental group of X by considering the monodromy of the system. The Riemann-Hilbert problem asks the converse: what systems of linear differential equations arise with prescribed monodromy representations? In this talk, I will discuss Deligne's solution to the Riemann-Hilbert problem for smooth, connected quasi-projective varieties over the complex numbers, following a survey by Nicholas Katz: https://web.math.princeton.edu/~nmk/old/DeligneXXIHilbert.pdf
Geometric and Abstract Variations of Hodge Structure
When: Fri, April 19, 2024 - 4:15pmWhere: Kirwan Hall 3206
Speaker: Myeong Jae Jeon (UMD) -
Abstract: Given an analytic family of smooth projective varieties over a complex manifold B, one can construct a holomorphic vector bundle over B whose fibers carry a polarized Hodge structure of weight k. This family of Hodge structures can be abstracted to the notion of a variation of Hodge structure (VHS) over B. In this talk, I will first describe the geometric VHS in the above setting and then define the abstract notion of VHS following E. Cattani's notes on VHS: https://webusers.imj-prg.fr/~fouad.elzein/Hodge.pdf
Geometric and Abstract Variations of Hodge Structure 2
When: Fri, April 26, 2024 - 4:15pmWhere: Kirwan Hall 3206
Speaker: Myeong Jae Jeon (UMD) -
Abstract: Given an analytic family of smooth projective varieties over a complex manifold B, one can construct a holomorphic vector bundle over B whose fibers carry a polarized Hodge structure of weight k. This family of Hodge structures can be abstracted to the notion of a variation of Hodge structure (VHS) over B. In this talk, I will first describe the geometric VHS in the above setting and then define the abstract notion of VHS following E. Cattani's notes on VHS: https://webusers.imj-prg.fr/~fouad.elzein/Hodge.pdf