Affine combinatorics, Schubert calculus, and total positivity

主要研究者：Lam，托马斯 提案编号：DMS - 0901111机构：仿射组合学，舒伯特演算，和全正性标题：哈佛大学PI提出发展源于仿射李代数和循环群的组合学。 PI与合作者将研究仿射李群的旗簇的舒伯特演算。 这包括理解舒伯特品种的几何和多项式的组合表示舒伯特品种（共）同源性和K-（共）同源性。 PI还将与同事一起为循环组及其标志品种开发一种总积极性理论。 正在开发的理论推广了经典的Edrei-Thoma分类的字符的无限对称群。 此外，出现了Coxeter群的新组合学，包括Coxeter群的极限元素的弱序。 在另一个方向上，PI与合作者将研究仿射Coxeter安排的凸几何，特别是在仿射Schubert簇和仿射Grassmannian的全正性研究中出现的某些多面体。PI的研究是在组合学领域，研究如何计算离散对象。PI研究几何学（研究空间物体的形状）和代数结构（研究多项式方程的解）中出现的组合问题。 PI工作的持续主题之一是研究数学中出现的（正）数。 这些数字可能会出现在计算几何物体在空间中相互作用的方式，或者计算多项式方程的某些解时。 特别是，PI旨在理解几何图形的“正”部分，就像正真实的轴是真实的线的“正”部分一样。 PI的工作将对理解几何、代数和组合学之间的关系产生重大影响。

ABSTRACTPrincipal Investigator: Lam, Thomas Proposal Number: DMS - 0901111 Institution: Affine combinatorics, Schubert calculus, and total positivityTitle: Harvard UniversityThe PI proposes to develop combinatorics arising from affine Lie algebras and loop groups. The PI with collaborators will study the Schubert calculus of flag varieties of affine Lie groups. This includes understanding the geometry of Schubert varieties and the combinatorics of the polynomials representing Schubert varieties in (co)homology and K-(co)homology. Together with co-workers, the PI will also develop a theory of total positivity for loop groups, and for their flag varieties. The theory being developed generalizes the classical Edrei-Thoma classification of characters of the infinite symmetric group. Furthermore, new combinatorics for Coxeter groups occurs, including a weak order for the limiting elements of a Coxeter group. In another direction, the PI with collaborators will study the convex geometry of the affine Coxeter arrangement, and in particular, certain polytopes which occur in the study of affine Schubert varieties and total positivity of the affine Grassmannian.The PI's research is in the area of combinatorics, which studies how to count discrete objects. The PI studies combinatorial problems which arise from geometry (studying shapes of objects in space) and algebraic structures (studying solutions to polynomial equations). One of the on-going themes of the PI's work is the study of (positive) numbers which arise in mathematics. These numbers may occur when counting the ways geometrical objects interact in space, or by couting certain solutions to polynomial equations. In particular, the PI aims to understand the "positive" part of a geometrical figure in the same way the positive real axis is the "positive" part of the real line. The PI's work will have a significant impact on the understanding of the relationships between geometry, algebra, and combinatorics.