Combinatorics of affine Schubert calculus, K-theory, and Macdonald polynomials

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This project concerns the study of missing core ideas in the theory of k-Schur functions connected to Macdonald polynomials, Gromov-Witten invariants, the co(homology) and K-theory of affine Grassmannians, and the WZW model of conformal field theories. The k-Schur functions are symmetric functions discovered in a study of Macdonald polynomials. Pursuant work with k-Schur functions has led to a theory that branches into many fields along the lines of modern Schubert calculus. Classical Schubert calculus addressed enumerative problems in projective geometry and used intersection theory to convert them into problems of computation in the cohomology ring of the Grassmannian. The explicit realization of such computations using combinatorial Schur theory played a major role in transforming Schubert calculus into a contemporary theory that provides elegant solutions to a diverse body of problems. In a similar manner, the introduction of k-Schur functions has generated a pool of open problems in geometry, representation theory, algebra, and physics. Combinatorics is a vast area of mathematics loosely described as the study of counting collections of objects that satisfy specified criteria. As such, combinatorics plays an integral role in the development of many fields, and combinatorial methods are employed by scientists ranging from biologists to theoretical physicists. The PI studies a refinement for the combinatorics of symmetric functions, a classical part of mathematics with applications in fields including physics, engineering, and computer science.