Collaborative Research: Cohomology, Deformations, and Invariants

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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.