2023 Fall

Title: Learning seminar on geometry and topology problems related to arithematic variety

Organizer: Shan Tai Chan, Siqi He, Jie Liu and Pengyu Yang

Introduction: Superrigidity is a concept designed to demonstrate how a linear representation ρ of a discrete group within an algebraic group G can, under certain circumstances, be as effective as a representation of G itself. The celebrated works of Margulis have addressed superrigidity in arithmetic varieties with a rank greater than one.

In this learning seminar, our goal is to delve into recent progress on geometry and topology problems associated with arithmetic varieties, particularly focusing on the rigidity problems that extend beyond Margulis' contributions. Specifically, we aim to explore the work of Koziarz-Maubon [1] on the Toledo invariant of representations of complex hyperbolic lattices, the research by Bader-Fisher-Miller-Stover [2] [3] on arithmeticity and superrigidity, and the proof of the Ax-Schanuel conjecture presented by Mok-Pila-Tsimerman [4].

References:

[1] V.Koziarz, J.Maubon. Maximal representations of uniform complex hyperbolic lattices. Ann. of Math. (2), 185(2):493–540, 2017.

[2] U.Bader, D.Fisher, N.Miller, M.Stover, Arithmeticity, superrigidity, and totally geodesic submanifolds Ann. of Math. (2)193(2021), no.3, 837–861.;

[3] U.Bader, D.Fisher, N.Miller, M.Stover, Arithmeticicity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds, Invent. Math.233(2023), no.1, 169–222.

[4] N.Mok, J. Pila, J.Tsimerman. Ax-Schanuel for Shimura varieties, Ann. of Math. Pages 945-978 from Volume 189 (2019).