Restricted Research - Award List, Note/Discussion Page
Fiscal Year: 2023
264 University of North Texas (142152)
Principal Investigator: Shepler,Anne V
Total Amount of Contract, Award, or Gift (Annual before 2011): $ 5,000
Exceeds $250,000 (Is it flagged?): No
Start and End Dates: 9/1/22 - 8/31/27
Restricted Research: YES
Academic Discipline: Mathematics
Department, Center, School, or Institute: College of Science
Title of Contract, Award, or Gift: Twisted Tensor Products
Name of Granting or Contracting Agency/Entity:
Simons Foundation
CFDA:
Program Title: none
Note:
Recent work has concentrated on establishing conversion maps between practical resolutions (used concretely in applications) and the unwieldy bar resolution (which yields theoretical information) to obtain new results on noncommutative algebras. For example, Hochschild cohomology governs the deformations of algebras. The PIs work gives new results on deformations arising in combinatorics and representation theory like Lusztigs graded affine Hecke algebras, Etingof and Ginzburgs symplectic reflection algebras, and Drinfeld Hecke algebras. These new techniques in homological algebra give concrete conditions for PBW deformations arising from groups and Hopf algebras acting on polynomial rings, quantum polynomial rings, exterior algebras, and Koszul algebras in general. Collaborators of the PI on these projects include Sarah Witherspoon and Ellen Kirkman as well as PhD students. The PI also works on related conjectures in algebraic combinatorics on the invariant theory of reflection groups and hyperplane arrangements. Recent work gives generators for the set of differential derivations invariant under the action of a complex reflection groups and uncovers an unusual exterior algebra structure for well-generated (duality) reflection groups, including all Coxeter and Weyl groups. Results obtained for mixed derivation differential forms give applications to q-Narayana and q-Kirkman polynomials and h-vectors and f-vectors in combinatorics. Collaborators include Victor Reiner and Eric Sommers. Work with PhD students focuses on reflection groups acting in positive characteristic.
Discussion: No discussion notes