Recording: https://disk.pku.edu.cn:443/link/297EE2DDEAF407BA14690A9EC8850B8E
Valid Until: 2026-06-30 23:59
Abstract: The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this talk, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.
We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. Combined with the generalized Poincare conjecture established by Smale, Freedman and Perelman, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.
This talk is based on a joint work with Huanchen Bao.
Bio: Dr. Xuhua He is currently the Choh-Ming Li Professor of Mathematics at the Chinese University of Hong Kong (CUHK). Prof. He received his B.S. degree in mathematics from Peking University in 2001, and Ph.D. degree from Massachusetts Institute of Technology in 2005. Before joining CUHK, he used to work at the State University of New York at Stony Brook (2006-2008), Hong Kong University of Science and Technology (2008-2014), and the University of Maryland (2014-2019). Moreover, he was a von Neumann Fellow at the Institute for Advanced Study for the academic year 2016–2017, and a Simons Visiting Professor at the Université Sorbonne Paris Nord (Paris 13 University) in 2017. For his outstanding achievements in the fields of algebraic groups, representation theory, and arithmetic geometry, he was an invited speaker at the ICM (2018) and was awarded the Morningside Gold Medal of Mathematics in 2013 and the AMS Chevalley Prize in Lie Theory in 2022.
