Combinatorics in geometry and representation theory

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The PI proposes to work on problems relating combinatorics with geometry and representation theory. Together with co-workers, the PI aims to develop a theory of Schubert calculus on the affine Grassmannian from both a combinatorial and geometric perspective. This includes the study of affine Schubert polynomials in both homology and cohomology and associated formulae such as Pieri rules. In the combinatorial side, affine generalizations of classical algorithms such as Schensted insertion will be studied. On the geometric side, the PI will focus on understanding different positivity properties geometrically and to connect the affine Grassmannian with Macdonald polynomials. Another part of the PI's work will focus on questions related to Schur positivity. The PI with collaborators have recently resolved several open Schur positivity problems that attracted a lot of attention, including conjectures of Fomin-Fulton-Li-Poon and of Okounkov. They plan to apply their techniques to prove several other prominent conjectures concerning Schur positivity. In another direction, the PI with collaborators, aims to study the Kazhdan-Lusztig cells in type B with unequal parameters using the combinatorial algorithm known as domino insertion.The PI's research is in the area of combinatorics. Combinatorics is an area of mathematics concerned with counting and has received a lot of attention recently due to its applications to probability, cryptography and computer science. The PI's research is concerned with connecting the discrete (or finite) problems in combinatorics with complicated structures in geometry and algebra. In geometry, infinite and continuous objects are studied together with a notion of space and dimension. In algebra, symmetries are studied using systems of equations. The PI's research should reveal deep relationships between these areas and is likely to have an impact on all three fields.