CAREER: The cohomology and birational geometry of moduli spaces

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Homogeneous varieties, in particular, ordinary and isotropic Grassmannians, are central objects of study in algebraic geometry, representation theory, and combinatorics. The investigator proposes to develop positive algorithms for computing the structure constants of the cohomology of isotropic Grassmannians and isotropic flag varieties. In recent years, similar positive algorithms have led to the solutions of many important problems for Type A Grassmannians, including the saturation conjecture and the reality of Schubert calculus. The investigator will also study the effective cones of the moduli space of curves and stable maps. The moduli spaces of curves are among the most studied objects in mathematics. Their cones of effective divisors are important invariants, intimately tied to problems such as the Schottky problem, the Kodaira dimension of the moduli space of curves, and the existence of modular forms. Recently, understanding the effective cone of moduli spaces has received new impetus following the seminal work of Hacon, McKernan, and their collaborators on the Minimal Model Program (MMP). The investigator proposes to run MMP on moduli spaces such as the Kontsevich moduli spaces of genus zero stable maps or Hilbert scheme of points on the projective plane. Systems of polynomial equations occur in many facets of life ranging from evolutionary biology to physics and from cryptography to computer science. Algebraic geometry studies geometric properties of solutions of polynomial systems. The solutions that have many symmetries are especially interesting and important. For example, a sphere is a perfectly symmetric space in the sense that any point can be rotated to any other point. The investigator studies the number of solutions to polynomial systems involving such perfectly symmetric spaces called homogeneous varieties. The investigator also calculates more subtle geometric invariants of systems of polynomial equations such as how the behavior of the space of solutions changes under perturbations of the system.