Why does the mirror
reflect you left-right, and not up-down?Coxeter,
Ludwig Wittgenstein, and
AT UNT ALGEBRA SEMINAR PUBLICATIONSSpecial Session: "Noncommutative Algebra,
Deformations, and Hochschild Cohomology" Sarah Witherspoon and I
are organizing a Special Session conference
at the American Mathematical Society meeting
in Lubbockat Texas Tech
11--13, 2014. TORA = Texas-Oklahoma
Representations and Automorphic forms
I'm on the steering committee and organizer
for a conference
seriessupported by the National Science Foundation and
Oklahoma State University, University of Oklahoma, and University of
TORA I meeting: September 17--18, 2011, Univeristy of North
(Plenary speakers: Nils Skoruppa, Sol Friedberg, Martin Raum.)
IV meeting: March 23--24, 2013, University of North
(Plenary speakers: Ken Ribet, Nolan Wallach, Susie Kimport.)
My work combines
ideas in Algebra, Geometry, Invariant Theory, Representation Theory,
Recent interests include homological algebra,
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).
I'm also interested in group codes, i.e., codes in computer science
build on isometry groups and
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
mod out by symmetry to obtain the orbifold V/G which may have singularities.Geometrically, we
might replace V/G with a smooth variety,
cohomology recommends an algebraic approach: replace the ring of
invariant polynomials S^G with the
semi-direct product algebra S#G.
governs the deformation theory and predicts various
algebras important in representation theory, combinatorics, and the
geometry of orbifolds.
I have a special interest in reflection
groups. These 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), and
reflection groups over arbitrary fields. Their
study intertwines invariant theoryand arrangements of hyperplanes.
(Scott Crass can explain relations with Dynamical
My work has been supported by several
National Science Foundation:
Grant (DMS-1101177), Principal Investigator, 2011--2014
Grant (DMS-0800951), Principal
Research Grant (DMS 0402819), Principal
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: Young
Investigators Research Grant, Principal Investigator, 2002--2004 Alexander
von Humboldt Foundation: Research Fellowship (at RWTH
Aachen University), 2009
Texas Coordinating Board: Advanced
Research Program Grant ($43,469), PI, 2008--2010.
Masters/Ph.D. Advisor for: Paisa
(B.S. from MIT, M.S. from UNT, currently Ph.D.
student at Univ. of South Carolina),
Foster-Greenwood (B.S. from UNT, Ph.D. from UNT Summer 2012.) Christine Uhl
(Current Ph.D. student at UNT.) Philip Puente (Current
Ph.D. student at UNT. Tamara Knox (Current Ph.D.
student at UNT.)REU on
Group Coding Summer 2013 Research
experience for undergraduates on coding with isometry groups....
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.
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. GAP
You want Chevie with linux? No, you can't use gap4. Yes, you
must install gap3r4p4. Doesn't compile?
I attended the honors college at Valparaiso
University---a small, liberal arts school in Indiana.
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 a 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
Reflection groups and modular forms:
Images by Douglas Dunham (University of
Minnesota at Duluth), and Charlie Gunn with The Geometry Center
(University of Minnesota).
Coxeter says 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 vindication."