RUI: Affine Flags, p-adic Representations, and Quantum Cohomology

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This research project addresses problems in algebraic geometry, which studies solutions to systems of polynomial equations, and in representation theory, which aims to explain the basic building blocks of symmetry in mathematics and natural science. The central objects of study in this project are groups of invertible matrices with power series entries. Such algebraic groups over local fields have an especially beautiful decomposition into cells indexed by elements of a group of transformations that is generated by reflections across hyperplanes in Euclidean space. This cell decomposition permits an approach to understanding the algebraic geometry and representation theory of the matrix group by employing combinatorial and geometric techniques that exploit the abundant symmetry featured in the arrangement of the reflecting hyperplanes. The project also provides involves undergraduate students in mathematical research through summer research programs, year-long thesis projects, and participation in local colloquia, regional seminars, and national conferences.The investigator will utilize and extend surprising relationships among p-adic representation theory, affine flag varieties in positive characteristic, the quantum cohomology of complex Grassmannians, and the homology of the affine Grassmannian. Concrete goals of the research include explicit type-free formulas for dimensions of affine Deligne-Lusztig varieties, values of p-adic orbital integrals, and products of quantum and affine Schubert classes. The primary tool in most projects is the alcove walk model for the affine flag variety, a uniform combinatorial platform that connects the study of affine Hecke algebras, crystal bases, Mirkovic-Vilonen cycles, quantum and affine Schubert calculus, and geodesics in the building of Kac-Moody groups.