Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

表示论与代数几何中的多面体组合学研究课题：量子群及其表示的正则基参数化的热带微积分方法;全正簇与双Bruhat胞的几何;广义行列式微积分及其在半单群、Kac-Moody群及其量子化的正则基中的应用;广义舒伯特细胞在多个旗品种。该项目将几个数学学科联系在一起：代数几何，表示论，数学物理和多面体组合学。主要的统一成分是热带微积分，它起源于应用数学（控制理论和组合优化）。这表明了将数学视为一门独特学科的重要性，以及将传统上分离的思想和概念结合在一起的重要性。

Polyhedral Combinatorics in Representation Theory and Algebraic GeometryThe investigator studies the following research topics: tropical calculus approach to parametrizations of canonical bases for quantum groups and their representations; geometry of totally positive varieties and double Bruhat cells; generalized determinantal calculus with applications to canonical bases for semisimple groups, Kac-Moody groups and their quantizations; generalized Schubert cells in multiple flag varieties. The project ties together several mathematical disciplines: algebraic geometry, representation theory, mathematical physics, and polyhedral combinatorics. The main unifying ingredient is tropical calculus which takes its origin in applied mathematics (control theory and combinatorial optimization). This demonstrates the significance of viewing mathematics as a unique subject, and that of bringing together traditionally separated ideas and concepts.