Professor Shimozono studies combinatorial structures arising in representation theory and algebraic geometry. With Allen Knutson and Ezra Miller, he proved a conjecture of Buch and Fulton for the torus equivariant cohomology classes of degeneracy loci associated with an equioriented type A quiver, which generalizes the Giambelli-Thom-Porteous formula for determinantal varieties. He will study the equivariant K-theoretic and cohomology classes of degeneracy loci associated with other quivers. This research has applications to the Schubert calculus and the geometry of flag varieties. Shimozono will continue his research on combinatorial problems arising in the representation theory of classical and affine Lie algebras and Weyl groups. With Mike Zabrocki, he has discovered new creation operators for characters of classical type, which, together with crystal graph techniques, will be applied to the problem of finding explicit combinatorial formulae for affine Kazhdan-Lusztig polynomials. He also plans to continue his study of crystal bases of finite-dimensional modules over quantum affine algebras, with applications to branching and fusion multiplicities in conformal field theories and statistical mechanics.
Shimozono's research on quiver loci is related to some classical mathematics with current applications. For example, the cohomology classes of quiver loci may be viewed as polynomials which generalize the classical polynomials known as subresultants. Subresultants are used in computer algebra systems to efficiently compute the greatest common divisor of a collection of polynomials, and may also be used to find the number of common zeroes of a pair of polynomials without factoring them. His research on affine crystal bases involves the study of certain highly symmetrical graphs with colored edges. The symmetries exhibited by these graphs appear in many areas of mathematics and physics, such as in the low-temperature behavior of statistical mechanical models involving particles situated on a two-dimensional lattice.
