The PI proposes to develop combinatorics arising from affine Lie algebras and loop groups. The PI with collaborators will study the Schubert calculus of flag varieties of affine Lie groups. This includes understanding the geometry of Schubert varieties and the combinatorics of the polynomials representing Schubert varieties in (co)homology and K-(co)homology. Together with co-workers, the PI will also develop a theory of total positivity for loop groups, and for their flag varieties. The theory being developed generalizes the classical Edrei-Thoma classification of characters of the infinite symmetric group. Furthermore, new combinatorics for Coxeter groups occurs, including a weak order for the limiting elements of a Coxeter group. In another direction, the PI with collaborators will study the convex geometry of the affine Coxeter arrangement, and in particular, certain polytopes which occur in the study of affine Schubert varieties and total positivity of the affine Grassmannian.
The PI's research is in the area of combinatorics, which studies how to count discrete objects. The PI studies combinatorial problems which arise from geometry (studying shapes of objects in space) and algebraic structures (studying solutions to polynomial equations). One of the on-going themes of the PI's work is the study of (positive) numbers which arise in mathematics. These numbers may occur when counting the ways geometrical objects interact in space, or by couting certain solutions to polynomial equations. In particular, the PI aims to understand the "positive" part of a geometrical figure in the same way the positive real axis is the "positive" part of the real line. The PI's work will have a significant impact on the understanding of the relationships between geometry, algebra, and combinatorics.
