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

职业：组合代数几何：旗形簇、环面几何和应用

• 批准号: 2142656
• 负责人: Laura Escobar Vega
• 金额: $ 43.67万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142656&HistoricalAwards=false
• 关键词: CAREER; Combinatorial; Algebraic; Geometry; Flag

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.

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。对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取得了很大的进步，这方面有着很强的记录。这个项目研究和开发了组合数学和代数几何之间相互作用产生的离散结构。此外，组合学的跨学科潜力有两种利用方式。首先，使用最近的几何重述从机器学习中研究模型的复杂性。第二，开发一种算法，用于帮助识别影响人类健康和长寿结果的新的复杂遗传结构。同时，该项目的教育计划将增加那些传统上代表不足的人对数学的参与，从服务不足的社区招收有才华的学生参加数学博士项目，并传播现代组合学研究。更明确地说，研究部分的目标是促进旗型变种的组合子变种(即Hessenberg变型和Kazhdan-Lusztig变型)的理论，提出牛顿-奥库科夫体的对偶概念，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，以及扩大未被充分代表的群体参与数学的活动。这项提议的教育部分包括华盛顿大学数学后联合学士学位课程的研究体验计划和哥伦比亚的组合学暑期学校ECCO。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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。