CAREER: Algebraic Combinatorics and its Applications

这份职业建议包括与代数和计数组合学有关的问题及其在多面体几何、表示理论、逆边界问题和代数几何中的应用。讨论了置换多面体及其推广，置换多面体是与Coxeter排列有关的某些凸多面体。PI提出了三种不同的方法来计算这种多面体的体积和Ehrhart多项式。他引入了具有显著组合性质的新的混合欧拉数。置换多面体的各种推广包括Stasheff结合面体、Pitman-Stanley多面体、与de Concini-Procesi的奇异紧化有关的多面体等。这些多面体与与Fomin-Zlevinsky的团簇代数理论有关的广义结合面体之间存在有趣的平行关系。命题的下一部分与舒尔的正性有关。PI和合作者最近解决了几个引起广泛关注的公开的Schur正性问题，包括Fomin-Fulton-Li-Poon猜想和Okounkov猜想。他们计划用他们的技术来证明其他几个突出的猜想。该提议提到了一个一般的Schur正性猜想，涉及一个神秘的多面体，它可能与Klyachko锥有关，并导致新的有趣的组合学。本文讨论了网络的边界逆问题及其与全正性的关系。网络将格拉斯曼曲线上的完全阳性细胞参数化。这一结构与Fomin-Zlevinsky关于双Bruhat胞格的结果和他们的簇代数理论有联系。这份职业建议书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的组合问题。这些问题涉及到对各种离散的数学对象进行计数或枚举，例如，在将复杂的几何对象分解为更简单的对象时，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)、表示理论(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。该协会计划鼓励研究生和本科生进行研究。组合问题对本科生很有吸引力，因为这些问题很直观，很容易表达，但却很有挑战性。该提案描述了在未来的组合学课程教学中使用交互式可视化工具和演示、小应用程序和动画的计划。

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.