does the mirror
reflect you left-right, and not up-down?Coxeter,
ALGEBRA SEMINARTORA = Texas-Oklahoma Representations and Automorphic forms
I'm on the steering committee and organizer for a new conference seriessupported
by the National Science Foundation
and Oklahoma State University,
University of Oklahoma, and University of North Texas:
TORA I meeting: September 17--18, 2011, Univeristy of North Texas. (Plenary speakers: Nils Skoruppa, Sol Friedberg, Martin Raum.)
TORA IV meeting: March 23--24, 2013, University of North Texas. (Plenary speakers: Ken Ribet, Nolan Wallach, Susie Kimport.)
Special Session: "Combinatorial Avenues in Representation Theory" Nathaniel Thiem, Richard Green, and I are organizing a Special Session conference at the American Mathematical Society meeting in Boulder, Colorado, April 13--14, 2013.
My work combines ideas in
Geometry, Invariant Theory, Representation Theory, and Combinatorics.
I've been working in homological
algebra, examining deformation theory, cohomology, and Drinfeld
Orbifold Algebras (which include symplectic reflection algebras, rational Cherednik
algebras, graded Hecke algebras, Drinfeld Hecke algebras, Weyl
algebras, universal enveloping algebras, and twists by a group action
or quantum parameters).
Why? Physicists often regard space as a
Calabi-Yau manifold endowed with symmetry. We model the local setting
with a finite group G acting linearily on a finite dimensional vector
space V. We mod out by the symmetry and
regard the orbifold V/G. The coordinate ring is the
of invariant polynomials k[V*]^G. The orbifold V/G may have singularities
and one seeks geometrically to replace V/G with a smooth variety. Hochschild cohomology recommends an algebraic approach: replace the ring of invariant polynomials with the
natural semi-direct product algebra k[V^*]#G (a skew group algebra).
cohomology governs the deformation theory of k[V^*]#G and predicts
various algebras important in representation theory, combinatorics, and
geometry of orbifolds.
I have a special interest in reflection
are groups (acting on a finite dimensional vector space) generated
by reflections: elements that fix a hyperplane (or "mirror")
pointwise. They include
the Weyl and Coxeter groups, complex reflection groups (u.g.g.r.'s),
reflection groups over arbitrary fields. Topics include
invariant theory, arrangements
of hyperplanes, regular polytopes, Hecke algebras, coinvariant
algebras, Coxeter groups,
Groups, and Braid groups.
(Scott Crass can explain relations with Dynamical
work has been supported by
Research Grant (DMS-1101177),
Principal Investigator, 2011--2014
Research Grant (DMS-0800951),
Principal Investigator, 2008--2011
0402819), Principal Investigator, 2004--2008
N.S.F. Post-Doctoral Research Fellowship (Award
9971099), Principal Investigator, 1999--2002
TORA Conference Grant (DMS-1132586), Co-Principal Investigator, 2011--2012
TORA Conference Grant (DMS-1302770), Co-Principal Investigator, 2013--2014
National Security Agency:
Investigators Research Grant, Principal Investigator, 2002--2004 Alexander von Humboldt
Foundation: Research Fellowship (at RWTH Aachen
Texas Coordinating Board:
Research Program Grant ($43,469), PI, 2008--2010.
PUBLICATIONSTALKS GRADUATE STUDENTS Masters/Ph.D.
Advisor for: Paisa
MIT, M.S. from UNT, currently Ph.D. student at Univ. of South
Briana Foster-Greenwood (B.S. from
UNT, Ph.D. from UNT expected summer 2012.) Christine Uhl
(Current Ph.D. student at UNT.) Workshop on Algebras Joint workshop
on algebras, deformation theory, representation theory for and by
postdocs/Ph.D. students from Texas A&M University, Baylor
University, University of North Texas. Conference on Hecke Algebras Matt Douglass and I are organizing a
special session on Hecke
Algebras and Deformations in Geometry and Topology at the AMS
meeting in St. Paul, Minnesota, April 10-11, 2010.
You want Chevie with
linux? No, you can't use gap4. Yes, you must
install gap3r4p4. Doesn't
I attended the honors college
University---a small, liberal arts school in
I minored in the humanities, co-founded a comedy troupe, participated
in many theatre productions, and worked for the music department as a
piano accompanist. I decided to major in math after
participating in an Research Experience for Undergraduates program at the University
of Oklahoma. I also spent a semester at Hangzhou
University in China (took Chinese language classes
and also taught English at the Y.M.C.A.).
Afterwards, I moved to California for grad school and scuba diving. Moray eels provide nice examples for constructing orbifolds. Maybe not wolf
Douglas Dunham (University of Minnesota at Duluth), and
Gunn with The Geometry Center (University of Minnesota).
of Escher's print: "He got it absolutely right to the
millimetre, absolutely to the millimetre. ... Unfortunately, he didn't
live long enough to see my mathematical