中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

学术报告

报告人： 鲍涣辰 教授 （新加坡国立大学）
题  目：Regularity theorem for totally non-negative flag varieties
时  间：2022.10.27（星期四），14:30-15:30
地  点：数学院南楼N802

Zoom会议：466 356 2952   密码：mcm1234      
摘  要：The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this talk, I will introduce a (new) J-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.TheJ-totally nonnegative flag variety enjoys many favourable properties like the ordinary totally nonnegative flag variety. Moreover, the J-total positivity on the full flag variety provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, 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 is based on joint work with Xuhua He. This is the continuation of the previous talk. However, I will recall the basics at the beginning. 
-------------------------

报告人： 鲍涣辰 教授 （新加坡国立大学）
题  目：Frobenius splittings via quantum symmetric pairs
时  间：2022.11.03（星期四），14:30-15:30
地  点：数学院南楼N802

Zoom会议：466 356 2952   密码：mcm1234      
摘  要：Let G be a connected reductive group over an algebraically closed field of characteristic not 2. Let \theta be a (quasi-split) involution of G with the fixed point subgroup denoted by K. Using the theory of i-canonical bases on quantum symmetric pairs, we construct an explicit Frobenius splitting for the symmetric subgroup K and Frobenius splittings for K-orbits on the flag variety.Along the way, we construct a symmetric subgroup scheme over Z[2^{-1}] with geometric fibre K, generalizing Lusztig's construction of Chevalley group schemes using quantum groups. This is joint work in progress with Jinfeng Song (NUS).