CAREER: Hessenberg Varieties, Symmetric Functions, and Combinatorial Representation Theory

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Research in algebraic combinatorics seeks to build connections between discrete structures and algebraic objects, with broad applications in mathematics and other sciences. This project develops tools to organize discrete data in a way that reflects key structural properties. These tools are applied to study solutions of complicated systems of polynomial equations and streamline computation in order to decipher patterns in otherwise complex data. The resulting insights yield new approaches to important unsolved problems in both geometry and combinatorics. The educational component of this project will train the next generation of scientists through a targeted mentoring program that includes research opportunities for traditionally underrepresented students. It also creates structured pathways for outreach to students in local schools.The research component of this project is concerned with the combinatorial and geometric structure of Hessenberg varieties. Hessenberg varieties are subvarieties of the flag variety whose cohomology rings encode rich combinatorial structure. Topological data obtained from these varieties will be used to make new in-roads toward open positivity conjectures in algebraic combinatorics and their extension to arbitrary Weyl groups. The geometry of Hessenberg varieties is completely understood in only a few cases. The PI will transform the traditional path of research in this area by studying families of Hessenberg varieties united by only a few essential properties, and prove the geometry of these varieties fluctuates in predictable ways as other inputs vary. This project also expands an existing undergraduate research program at Washington University in St. Louis to include students whose high school background does not prepare them to take advanced courses quickly. Funds will be used to establish a new math outreach seminar.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.