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.
This project tied together combinatorial problems from geometry, representation theory, physics, and computation. Combinatorics is an active and central branch of pure and applied mathematics. Because the field is concerned with the development of tools for analyzing, organizing, and arranging discrete data, combinatorial methods are rudimentary in many scientific areas such as genomics, computer science, statistics, and physics. The methods can often be traced back to research inspired by problems in algebra and geometry. For example, the RSA public-key encryption algorithm is based on an elementary combinatorial result in modular arithmetic. The outcome of this project is an artillery of combinatorial tools for attacking problems that connect to algebraic and geometric areas such as symmetric function theory, an area with applications to probability and statistical mechanics. Among the results are answers to questions such as "how many lines are there satisfying a number of generic intersection conditions?", "how many elements are in given sets and what properties do these sets have?", "how do fields correlate?", and "are there efficient algorithms for calculating these numbers or objects?" The investigation was fueled by extensive computational experimentation. A mutually beneficial component was the further development of the open-source SAGE software package, impacting the mathematics, physics, and computer science communities.
