Abstract：
An invertible matrix is called totally positive if all its minors are positive. In 1994, Lusztig initiated a far-reaching generalization of this classical notion to arbitrary split real reductive groups and their flag varieties, revealing deep connections between combinatorics, geometry, and representation theory. Since then, total positivity has found striking applications in diverse areas including cluster algebras, higher Teichmüller theory, and the physics of amplituhedra.
In this talk, I will survey the basic theory and highlight recent advances in total positivity. I will explain how total positivity intertwines with combinatorial structures such as the Bruhat order and the shellability of certain posets, with representation theory through Lusztig's canonical basis, and with topology via the topology of regular CW complexes and the role of the Poincaré conjecture.
I will discuss the recent joint work with Huanchen Bao and with Kaitao Xie. We show that the totally nonnegative flag varieties and, more recently, the links of double Bruhat cells in Kac-Moody groups are regular CW complexes. These results settle long-standing conjectures, including those of Björner (1984) on shellability, Fomin-Zelevinsky (2000) on double Bruhat cells, and further conjectures of Postnikov, Galashin-Karp-Lam, and Williams.
The talk aims to be accessible to a broad mathematical audience, highlighting the rich interplay between algebra, geometry, and combinatorics that makes total positivity such a vibrant area of research.
报告人简介：
何旭华，香港大学讲座教授、新基石研究员、香港科学院院士、香港数学会理事长。曾任美国马里兰大学数学系正教授、香港中文大学卓敏数学讲座教授，并曾担任普林斯顿高等研究院冯·诺依曼Fellow。研究方向涵盖算术几何、代数群与表示论。因在上述领域的重要贡献，屡获殊荣：2013年荣获晨兴数学金奖，2020年获科学探索奖，2022年获美国数学会Chevalley奖，并于2025年当选美国数学会会士。曾受邀在2015年“当代数学进展”会议上作报告，2018年成为国际数学家大会45分钟邀请报告人。此外，他还担任了2022年国际数学家大会李理论方向的核心选委。
