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

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.