报告题目：Total positivity: combinatorics, geometry, and representation theory

报告人：何旭华（香港大学讲座教授）

报告时间：2026年3月24日下午16：00

报告地点：格物楼数学研究中心528报告厅

报告摘要：An invertible matrix is called totally positive if all of its minors are positive. Over the past several decades, this classical notion has evolved into a rich and far-reaching theory. In a foundational 1994 work, Lusztig extended total positivity to arbitrary split real reductive groups and their flag varieties, uncovering remarkable connections with combinatorics, geometry, and representation theory. Since then, total positivity has emerged in a wide range of contexts, from cluster algebras and higher Teichmuller theory to the geometry underlying the amplituhedron.

In this talk, I will give an overview of the basic theory of total positivity and describe some recent developments. I will explain how the subject is intertwined with combinatorial structures such as Bruhat order and shellable posets, with representation theory through Lusztig’s canonical basis, and with topology through the study of regular CW complexes and with an unexpected appearance of the Poincare conjecture. I will also discuss recent progress on the conjectures of Bjorner, of Fomin-Zelevinsky, as well as further conjectures of Postnikov, Galashin-Karp-Lam, and Williams. This talk is based on my joint works with Huanchen Bao and with Kaitao Xie.

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。