Combinatorics in Representation Theory and Algebraic Geometry

Shimozono教授研究表示论和代数几何中出现的组合结构。他与Allen Knutson和Ezra Miller一起证明了BuchandFulton关于退化轨迹的环面等变上同调类的一个猜想，该猜想推广了行列式簇的Giambeli-Thom-Porteous公式。他将研究与其他箭图相关的退化轨迹的等变K-理论和上同调类。这项研究在Schubert演算和旗簇的几何中有应用。Shimozono将继续研究经典李代数、仿射李代数和Weyl群表示理论中出现的组合问题。他和Mike Zabrocki一起发现了经典类型特征的新的创造算子，这些算子将与晶图技术一起应用于寻找仿射Kazhdan-Lusztig多项式的显式组合公式的问题。他还计划继续他对量子仿射代数上内维模的晶基的研究，应用于形式场理论和统计力学中的分支和融合多重性。Shimozono对箭点轨迹的研究与一些经典数学有关，目前应用广泛。例如，箭图轨迹的上同调类可以被视为多项式，它推广了被称为次结果的经典多项式。次结果在计算机代数系统中被用来有效地计算多项式集合的最大公因数，也可以用来求出一对多项式的公零点的个数，而不需要对它们进行因式分解。他对仿射晶基的研究涉及到对某些具有有色边的高度对称图的研究。这些图形所表现出的对称性出现在数学和物理的许多领域，例如在涉及二维晶格上的粒子的统计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.