Applications of Lie Theory: Combinatorial Algebraic Geometry and Symmetric Functions

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Algebraic combinatorics is an area of research that seeks to build connections between discrete structures and algebraic objects, with broad applications in computing, statistics, biology, and other subjects of mathematics. A central theme in combinatorics problems is to organize discrete data in a way that reflects key structural properties, thereby making it easier to analyze. This project applies the tools of algebraic combinatorics to study solutions of complicated systems of equations, called algebraic varieties. The PI will develop sophisticated counting techniques to streamline computations and decipher patterns in otherwise complex data. The PI then will use geometric properties of algebraic varieties to uncover new approaches to unsolved problems in algebra and combinatorics. In addition this project also provides research training opportunities for graduate students.The specific research addressed in this project concerns the combinatorial and geometric structure of Hessenberg varieties and extended Springer fibers. Hessenberg varieties are subvarieties of the flag variety whose cohomology rings encode rich algebraic structure. The PI will use topological data obtained from Hessenberg varieties to outline a new approach to the long-standing Stanley-Stembridge conjecture in combinatorics. The geometry and topology of Hessenberg varieties is completely understood in only a few cases. Using combinatorial invariants and an affine paving, the PI will characterize geometric properties of Hessenberg varieties. Graham has defined an analogue of the Springer resolution, called the extended Springer resolution. The PI will use the fibers of this map to develop a new geometric framework for the generalized Springer correspondence, transforming the usual approach to this seminal work.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.