Affine combinatorics, Schubert calculus, and total positivity

主要研究人员：LAM，Thomas建议编号：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建议发展源于仿射李代数和环群的组合学。PI和合作者将研究仿射李群的FLAG簇的Schubert演算。这包括了解Schubert簇的几何，以及表示Schubert簇在(上)同调和K-(Co)同调中的多项式的组合。PI还将与同事一起开发环路群及其旗帜品种的全正性理论。正在发展的理论推广了经典的Edrei-Thoma关于无限对称群的特征标的分类。此外，出现了Coxeter群的新的组合学，包括Coxeter群的极限元素的弱序。在另一个方向，PI将与合作者一起研究仿射Coxeter排列的凸几何，特别是在仿射Schubert簇和仿射Grassmannian的全正性的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(研究空间中物体的形状)和代数结构(研究多项式方程的解)产生的组合问题。PI工作的一个持续主题是研究数学中出现的(正)数。当计算几何物体在空间中相互作用的方式时，或通过计算多项式方程的某些解时，可能会出现这些数字。特别是，PI的目的是理解几何图形的“正”部分，就像正实轴是实线的“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.