CAREER: Algebraic Combinatorics and its Applications

这个职业生涯的建议包括有关代数和枚举组合及其应用几何多面体，representationtheory，逆边界问题和代数几何的问题。 该命题讨论了置换多面体及其推广，它们是与Coxeter排列有关的凸多面体。 PI提出了三种不同的方法来计算这类多面体的体积和Ehrhart多项式，并引入了具有显著组合性质的混合欧拉数。 置换面体的各种推广包括Stasheff'sassociahedra，Pitman-Stanley多面体，与De Concini-Procesi的奇妙紧化相关的多面体等。这些多面体与Fomin-Zelevinsky的簇代数理论相关的广义关联面体之间有着有趣的相似之处。 下一部分的proposalis有关舒尔积极。 PI与合作者最近解决了几个开放的Schur正性问题，吸引了很多关注，包括Fomin-Fulton-Li-Poon猜想和Okounkov猜想。 他们计划应用他们的技术来证明其他几个突出的成就。 该建议提到了一个涉及神秘多面体的一般Schur正猜想，它可能与Klyachko锥有关，并导致新的有趣的组合数学。 本文讨论了网络的逆边值问题及其与全正性的关系。 网络参数化格拉斯曼上的全正细胞。 这种结构与Fomin-Zelevinsky的双Bruhat细胞的结果和他们的簇代数理论有联系。这个CAREER建议描述了组合学领域研究和教育的新举措。 组合技术在代数、几何、计算机科学、概率论、物理学、生物学、密码学等领域发挥着越来越重要的作用。该提案的研究和教育部分都针对组合学的应用。该提案讨论了现代数学研究前沿的几个重要组合问题。 这些问题涉及到对各种离散的数学对象进行计数或重新计数，例如，将复杂的几何对象分解为简单的对象时，多面体的顶点或碎片。 这些问题将有助于理解，澄清和简化非平凡的数学概念和结构。 拟议的研究项目将在几个领域产生影响，包括代数几何（使用代数研究几何对象），表示论（研究对称性），凸多面体理论和理论物理。 该提案的教育部分包括计划开设一门关于组合学现代应用的新课程。 PI计划鼓励研究生和本科生的研究。 组合问题对本科生很有吸引力，因为这些问题很直观，很容易公式化，但也很有挑战性。 该提案描述了在未来组合数学教学中使用交互式可视化工具、演示、小程序和动画的计划。

This CAREER proposal includes problems related to algebraic and enumerativecombinatorics and its applications to geometry of polytopes, representationtheory, inverse boundary problems, and algebraic geometry. The proposaldiscusses permutohedra and their generalizations, which are certain convexpolytopes related to Coxeter arrangements. The PI suggests three differentapproaches to calculation of volumes and Ehrhart polynomials of such polytopes.He introduces new mixed Eulerian numbers with remarkable combinatorialproperties. Various generalizations of permutohedra include Stasheff'sassociahedra, Pitman-Stanley polytopes, polytopes related to wonderfulcompactifications of De Concini-Procesi, etc. There are intriguing parallelsbetween these polytopes and generalized associahedra related toFomin-Zelevinsky's theory of cluster algebras. The next part of the proposalis related to Schur positivity. The PI with collaborators have recentlyresolved several open Schur positivity problems that attracted a lot ofattention, including Fomin-Fulton-Li-Poon's conjecture and Okounkov'sconjecture. They plan to apply their techniques to prove several otherprominent conjectures. The proposal mentions a general Schur positivityconjecture involving a mysterious polytope, which might have relations with theKlyachko cone and lead to new interesting combinatorics. The proposaldiscusses the inverse boundary problem for networks and its relations withtotal positivity. Networks parametrize the totally positive cells on theGrassmannian. This construction has links with Fomin-Zelevinsky's results ondouble Bruhat cells and with their theory of cluster algebras.This CAREER proposal describes new initiatives in research and education in thearea of combinatorics. Combinatorial techniques play an increasingly importantrole in other fields such as algebra, geometry, computer science, probabilitytheory, physics, biology, cryptography, etc. Both the research and theeducational components of the proposal aim on applications of combinatorics.The proposal discusses several important combinatorial problems on thefrontiers of modern mathematical research. These problems involve counting orenumerating various discrete mathematical objects, say, vertices of a polytopeor pieces in a decomposition of a complicated geometrical object into simplerobjects. These problems would help to understand, clarify, and simplifynontrivial mathematical concepts and constructions. The proposed researchproject would have impact in several fields, including algebraic geometry(which studies geometrical objects using algebra), representation theory (whichstudies symmetries), theory of convex polytopes, and theoretical physics. Theeducational part of the proposal includes plans for a new course on modernapplications of combinatorics. The PI is planning to encourage graduate andundergraduate research. Combinatorial problems are appealing for undergraduatestudents because these problems are intuitive and easy to formulate but yetquite challenging. The proposal describes plans for using interactive visualtools and demos, applets, and animations in teaching future combinatoricscourses.