CAREER: Combinatorial Algebraic Geometry: Flag Varieties, Toric Geometry, and Applications

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。离散结构的研究不仅在数学领域，而且在科学和工业领域都有很好的记录。这个项目研究和开发离散结构之间的相互作用所产生的组合学和代数几何。此外，组合学的跨学科潜力以两种方式利用。首先，使用最近的几何改写来研究机器学习模型的复杂性。第二，开发一种算法，用于帮助识别驱动人类健康和长寿结果的新的复杂遗传结构。与此同时，该项目的教育计划将增加那些传统上代表性不足的人对数学的参与，从服务不足的社区招募有才华的学生到博士。更明确地说，研究部分的目标是推进旗簇的组合子簇理论（即Hessenberg簇和Kazhdan-Lusztig簇），提出牛顿-Okounkov体的对偶概念，并研究广义置换面体在机器学习中的应用。该提案的更广泛影响包括有可能造福社会的跨学科项目和扩大数学代表性不足群体参与的活动。该提案的教育部分包括华盛顿大学数学联合学士后课程和哥伦比亚ECCO组合暑期学校的研究经验计划。该项目还将支持本科生和研究生的研究。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). There is a strong record of the study of discrete structures leading to advances not only within mathematics, but also in science and industry. This project studies and develops discrete structures arising from the interplay between combinatorics and algebraic geometry. Additionally, the interdisciplinary potential of combinatorics is utilized in two ways. First, to study the complexity of a model from machine learning using a recent geometric rephrasing. Second, to develop an algorithm used to help identify novel complex genetic structures driving health and longevity outcomes in humans. At the same time, the project’s educational program will increase the participation in mathematics of those traditionally underrepresented, recruit talented students from underserved communities to Ph.D. programs in mathematics, and disseminate modern research in combinatorics.More explicitly, the objectives of the research component are to advance the theory of combinatorial subvarieties of the flag variety (namely Hessenberg varieties and Kazhdan-Lusztig varieties), propose a notion of duality for Newton-Okounkov bodies, and study applications of generalized permutahedra to machine learning. The broader impacts of the proposal include interdisciplinary projects that have the potential to benefit society and activities that broaden the participation of underrepresented groups in mathematics. The educational component of this proposal consists of a research experiences program for Washington University’s Joint Post-Baccalaureate Program in Mathematics and the combinatorics summer school ECCO in Colombia. The project will also support research by undergraduate and graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.