The investigator will continue his study of quantum and arithmetic Schubert calculus, degeneracy loci, and related algebraic combinatorics. The following are some of the proposed research problems: a) complete his work on the classical and quantum Schubert calculus for isotropic Grassmannians and more general flag varieties; b) discover the right combinatorial objects to address the Giambelli problem in the quantum cohomology rings of these spaces; c) develop the related combinatorial theory of theta polynomials, a family of polynomials which intertwines the Schur S and Q functions; d) find a Littlewood-Richardson type rule for the product of two Schubert classes in the cohomology of isotropic Grassmannians. The educational activities proposed by the investigator include developing and teaching an undergraduate course based on problem solving with topics that have a direct connection to current research. He also will work on a book which aims to make the modern Schubert calculus and its various extensions and applications more accessible to beginners in the area.
This proposal is concerned with enumerative algebraic geometry and its interaction with other parts of mathematics such as Lie theory and combinatorics. The calculus invented by Schubert is a fundamental example of enumerative techniques in geometry which predate the cohomology and intersection theories of the twentieth century. An example of the kind of question considered by Schubert is: given 4 lines in Euclidean 3-space in general position, how many lines intersect all 4 of the given lines? Schubert calculus provides a general method to answer similar questions involving linear conditions in higher dimensions. The more recent definition of the quantum cohomology ring of a manifold incorporates invariants that count the number of rational curves in the space which satisfy natural incidence conditions. This theory has its origin in quantum physics, where it has applications to string theory and the predictions of mirror symmetry. The homogeneous spaces of Lie groups and their Schubert varieties provide a rich source of examples where explicit computations of quantum cohomology rings are possible. The resulting modern Schubert calculus is the focus of this research project. Besides the many applications to related areas such as equivariant cohomology and quiver varieties, it introduces fresh new ways of looking at rather classical objects, and leads to important combinatorial and computational challenges.
