Collaborative Research: Cohomology, Deformations, and Invariants

Shepler和Witherspoon将发展一种变形理论，将其推广到分次Hecke代数。他们将研究结合对称群和函数代数的斜群代数的变形。在表示论和非对易几何中，许多杰出的数学家在工作中独立地出现了这些代数的特殊变形，但仍有许多悬而未决的问题。谢普勒和威瑟斯彭将使用不变量理论、组合学、同调代数和表示理论的混合方法创建的新工具来回答其中一些问题。他们还将解决一些关于Hopf代数的结构和上同调的基本公开问题，并证明关于模反射群、不变理论和超平面排列的几个猜想。自然界中的物体通过它们的对称性来展示自己，例如晶体、分子、DNA和量子系统。当我们使一个物体变形时，我们改变甚至破坏了对称性。值得注意的是，我们经常在研究物体的变形后发现它的新属性。Shepler和Witherspoon关于分次Hecke代数及其相关变形的研究计划涉及各种数学领域，并源于数学界对分次Hecke代数表现出的爆炸性兴趣。Hecke代数在整个数学中无处不在，出现在代数、几何、数论、组合学、拓扑学、统计学、调和分析、数学物理、特殊函数、量子群、纽结理论和共形场论中。谢普勒和威瑟斯彭活跃在数学界，指导学生和博士后，与国际专家合作，组织会议和研讨会。他们的研究计划支持这些更广泛的活动。

Shepler and Witherspoon will develop a theory of deformations expanding that for graded Hecke algebras. They will study deformations of skew group algebras that combine groups of symmetries with algebras of functions. Particular deformations of these algebras arose independently in work by many prominent mathematicians in representation theory and noncommutative geometry, but many open questions remain. Shepler and Witherspoon will answer some of these questions using new tools created by blending methods from invariant theory, combinatorics, homological algebra, and representation theory. They also will solve some basic open problems about the structure and cohomology of Hopf algebras and prove several conjectures on modular reflection groups, invariant theory, and arrangements of hyperplanes. Objects throughout the natural world reveal themselves through their symmetries, for example, crystals, molecules, DNA, and quantum systems. When we deform an object, we alter or even break the symmetry. Remarkably often, we discover new attributes of the object after studying its deformations. Shepler and Witherspoon's research program on graded Hecke algebras and related deformations addresses a variety of mathematical fields and grows from the exploding interest the mathematical community shows in graded Hecke algebras. Hecke algebras are pervasive throughout mathematics, appearing in algebra, geometry, number theory, combinatorics, topology, statistics, harmonic analysis, mathematical physics, special functions, quantum groups, knot theory, and conformal field theory. Shepler and Witherspoon are active in the mathematical community, mentoring students and postdocs, collaborating with international experts, and organizing conferences and workshops. Their research program supports these broader activities.