Puzzles, Quantum K-Theory, and Other Topics in Schubert Calculus

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This research project concerns several open questions in the general area of enumerative geometry. The motivating goal in this subject is to determine the number of geometric figures of a fixed type that satisfy a list of conditions. These conditions could involve that the figures contain given points, meet other figures, or have tangents with specified properties. Surprisingly, while it can be extremely difficult to determine the precise list of solution figures that satisfy the conditions, it is often possible to predict the number of such solutions. The search for formulas for the number of solutions to enumerative problems has uncovered deep and beautiful combinatorial structures that are of interest in numerous fields, including geometry, combinatorics, representation theory, complexity theory in computer science, and mirror symmetry in physics. The PI will engage both graduate and undergraduate students in his research with the aim of recruiting new talent to mathematics in general and to enumerative geometry in particular. The PI is currently the thesis advisor of one Ph.D student, and two additional students have recently completed their Ph.D degrees under his supervision. The PI has supervised projects in the Research Experience for Undergraduates (REU) program at Rutgers University in 7 out of the past 8 summers. The awarded research grant will enable him to continue this activity. The PI will also develop computer software to facilitate research in his area, and he will help organizing conferences and workshops.Much information about the enumerative geometry of subvarieties in a flag manifold is encoded in its cohomology ring, as well as in more general cohomology theories. A major goal in Schubert calculus is to obtain formulas for the multiplicative structure constants of this ring with respect to its basis of Schubert classes. The PI will attempt to prove a positive combinatorial formula that expresses the cohomological structure constants of any three-step flag variety as the number of puzzles that can be created using a list of puzzle pieces. The Gromov-Witten invariants of a flag manifold X count the number of rational curves that meet general Schubert varieties in X when this number is finite. When infinitely many curves meet a general configuration of Schubert varieties, the collection of these curves form a moduli space called a Gromov-Witten variety. Gromov-Witten varieties in turn define K-theoretic Gromov-Witten invariants that are encoded in the quantum K-theory ring of X. The study of this ring provides a useful benchmark for our understanding of the singularities and rationality properties of Gromov-Witten varieties and related spaces. The PI will attempt to answer several open questions about positivity and finiteness properties of quantum K-theory. The methods to be used in this project come from geometry and combinatorics, with computer experiments as a central tool.