Combinatorics in geometry and representation theory

PI建议研究与几何和表示论有关的组合数学问题。 与同事一起，PI的目标是从组合和几何的角度发展仿射格拉斯曼的舒伯特微积分理论。 这包括研究仿射舒伯特多项式在同调和上同调和相关的公式，如皮耶里规则。 在组合方面，仿射推广的经典算法，如Schensted插入将进行研究。 在几何方面，PI将专注于 几何上理解不同的正性性质，并将仿射格拉斯曼多项式与麦克唐纳多项式联系起来。 PI的另一部分工作将集中在与舒尔积极性相关的问题上。PI与合作者最近解决了几个开放的舒尔正性问题，吸引了很多关注，包括Fomin-Fulton-Li-Poon和Okounkov的论文。他们计划应用他们的技术来证明其他几个关于舒尔积极性的突出论点。 在另一个方向，PI与合作者，旨在研究Kazhdan-Lusztig细胞类型B与不等参数使用组合算法称为多米诺插入. PI的研究是在组合学领域. 组合数学是一个与计数有关的数学领域，由于其在概率、密码学和计算机科学中的应用，最近受到了很多关注。 PI的研究涉及将组合学中的离散（或有限）问题与几何和代数中的复杂结构联系起来。 在几何学中，无限和连续的物体与空间和维度的概念一起研究。 在代数中，对称性是用方程组来研究的。 PI的研究应该揭示这些领域之间的深层关系，并可能对所有三个领域产生影响。

The PI proposes to work on problems relating combinatorics with geometry and representation theory. Together with co-workers, the PI aims to develop a theory of Schubert calculus on the affine Grassmannian from both a combinatorial and geometric perspective. This includes the study of affine Schubert polynomials in both homology and cohomology and associated formulae such as Pieri rules. In the combinatorial side, affine generalizations of classical algorithms such as Schensted insertion will be studied. On the geometric side, the PI will focus on understanding different positivity properties geometrically and to connect the affine Grassmannian with Macdonald polynomials. Another part of the PI's work will focus on questions related to Schur positivity. The PI with collaborators have recently resolved several open Schur positivity problems that attracted a lot of attention, including conjectures of Fomin-Fulton-Li-Poon and of Okounkov. They plan to apply their techniques to prove several other prominent conjectures concerning Schur positivity. In another direction, the PI with collaborators, aims to study the Kazhdan-Lusztig cells in type B with unequal parameters using the combinatorial algorithm known as domino insertion.The PI's research is in the area of combinatorics. Combinatorics is an area of mathematics concerned with counting and has received a lot of attention recently due to its applications to probability, cryptography and computer science. The PI's research is concerned with connecting the discrete (or finite) problems in combinatorics with complicated structures in geometry and algebra. In geometry, infinite and continuous objects are studied together with a notion of space and dimension. In algebra, symmetries are studied using systems of equations. The PI's research should reveal deep relationships between these areas and is likely to have an impact on all three fields.