Geometry, Arithmetic, and Combinatorics of Schubert Calculus

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The investigator will study the generalization of the classical Schubert calculus to the homogeneous spaces of Lie groups and related applications to enumerative algebraic geometry, arithmetic geometry, quantum cohomology, and algebraic combinatorics. The proposed research problems include the following: (a) Formulate and prove a rule for computing the product of two Schubert classes in the cohomology of flag manifolds; (b) establish a Pieri type rule in the cohomology and quantum cohomology ring of homogeneous spaces; (c) find a good theory of quantum Schubert polynomials for the orthogonal and symplectic Lie groups; (d) study the combinatorial theory of theta and eta polynomials, in particular their double versions, with applications to equivariant cohomology; (e) achieve more efficient computations in the Arakelov geometry of flag varieties. The educational activities proposed by the investigator include incorporating problems stemming from his research into the University of Maryland High School Mathematics Competition. He also is working on a book on Schubert calculus and its various modern extensions which will emphasize the geometric aspects of the theory, in a uniform manner across the different Lie types.In the late 19th century, Hermann Schubert made a first systematic study of enumerative projective geometry and invented a calculus which enabled him to solve a plethora of enumerative problems in a systematic fashion. An example of the kind of question he addressed is: given four randomly generated lines in Euclidean 3-space, how many lines will intersect all four of the given lines? Schubert's fundamental work was a precursor of the cohomology and intersection theories of the 20th century, and their modern extensions today, with applications to quantum physics (via string theory) and number theory (via arithmetic intersection theory). Moreover, the entire story can be placed in a more symmetric framework, and told using the language of Lie groups and their representation theory, at an increasing level of combinatorial complexity and importance. The investigator has studied these theories over a period of many years and has shown how to achieve combinatorially explicit computations and formulas which give a global picture of the ring structure in each case. Recently, he has succeeded in doing this in a uniform way across all classical Lie types -- in the case of cohomology -- and has thus opened new avenues of research in the rapidly growing field of modern Schubert calculus.This award is co-funded by the Combinatorics Program.