Combinatorics in Representation Theory and Algebraic Geometry

教授Shimozono研究组合结构所产生的表象理论和代数几何。与Allen Knutson和Ezra米勒一起，他证明了Buchand富尔顿关于与等向型A型环相关的退化轨迹的环面等变上同调类的猜想，推广了行列式簇的Giambelli-Thom-Porteous公式。他将研究等变K理论和上同调类的退化轨迹与其他quivers。这一研究对Schubert演算和旗簇的几何有应用。Shimozono将继续他的研究组合问题所产生的代表性理论的经典和仿射李代数和Weyl群。与迈克Zabrocki，他发现了新的创造运营商的字符的经典类型，这与晶体图技术，将被应用于问题的发现明确的组合公式的仿射Kazhdan-Lusztig多项式。他还计划继续研究量子仿射代数上的有限维模的晶体基，并将其应用于共形场论和统计力学中的分支和融合多重性。例如，上同调类的多项式可以被看作是推广了经典多项式的子结式。子结式在计算机代数系统中被用来有效地计算一组多项式的最大公约数，也可以用来找到一对多项式的公共零点的数量，而不需要因式分解它们。他对仿射晶体基地的研究涉及到某些高度对称的有色边图的研究。这些图所展示的对称性出现在数学和物理学的许多领域，例如在涉及二维晶格上粒子的物理力学模型的低温行为中。

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.