Combinatorics in Representation Theory and Algebraic Geometry

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Professor Shimozono studies combinatorial structures arising inrepresentation theory and algebraic geometry. WithAllen Knutson and Ezra Miller, he proved a conjecture of Buchand Fulton for the torus equivariant cohomology classes ofdegeneracy loci associated with an equioriented type A quiver,which generalizes the Giambelli-Thom-Porteous formula fordeterminantal varieties. He will study the equivariant K-theoreticand cohomology classes of degeneracy loci associated with otherquivers. This research has applications to the Schubert calculusand the geometry of flag varieties. Shimozono will continue hisresearch on combinatorial problems arising in the representationtheory of classical and affine Lie algebras and Weyl groups. WithMike Zabrocki, he has discovered new creation operators forcharacters of classical type, which, together with crystal graphtechniques, will be applied to the problem of finding explicitcombinatorial formulae for affine Kazhdan-Lusztig polynomials. Healso plans to continue his study of crystal bases offinite-dimensional modules over quantum affine algebras, withapplications to branching and fusion multiplicities in conformalfield theories and statistical mechanics.Shimozono's research on quiver loci is related to some classicalmathematics with current applications. For example, the cohomologyclasses of quiver loci may be viewed as polynomials whichgeneralize the classical polynomials known as subresultants.Subresultants are used in computer algebra systems to efficientlycompute the greatest common divisor of a collection ofpolynomials, and may also be used to find the number of commonzeroes of a pair of polynomials without factoring them. Hisresearch on affine crystal bases involves the study of certainhighly symmetrical graphs with colored edges. The symmetriesexhibited by these graphs appear in many areas of mathematics andphysics, such as in the low-temperature behavior of statisticalmechanical models involving particles situated on atwo-dimensional lattice.