Representation Theory and Moduli Spaces via Young Tableaux and Parking Functions

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This research focuses on the use of combinatorial tools, such as orderings of numbers or algorithms for parking cars, to provide a more concrete understanding of abstract mathematical concepts in geometry and algebra. In modern-day Schubert calculus, many questions in enumerative geometry - counting intersections of lines, planes, curves - have been translated into discrete combinatorial problems or algorithms that a computer can then analyze. The aim of this project is to come up with these types of combinatorial rules in related areas of geometry and algebra, in order to both increase computational efficiency and to make the geometric constructions more accessible to scientists in other disciplines. The geometric spaces and algebraic structures that will be studied are of central importance to quantum physics and string theory. The project will involve graduate students in the research.This work will particularly focus on subvarieties and generalizations of flag varieties, Grassmannians, and moduli spaces of curves, all three of which are important geometric spaces whose cohomology rings are graded S_n-modules. We aim to give combinatorial rules, in terms of Young tableaux, parking functions, and other combinatorial objects, that govern computational aspects of their cohomology rings, and use them to resolve open questions about the corresponding geometric spaces. These new combinatorial rules also will be used to approach long-standing open problems in symmetric function theory, including the Macdonald positivity conjecture and the problem of determining equality of skew Schur Q functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.