Why does the mirror
reflect you left-right, and not up-down?Coxeter,
Ludwig Wittgenstein, and
E.C. Escher....
Fibonacci Numbers: Oanh Nguyen
was fast with the Kroger flower and her ipad: SpiralsOneWay,SpiralsOtherWay.
ALGEBRA
SYMPOSIUM 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
University, April
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
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.)
MATHEMATICAL RESEARCH
My work combines
ideas in Algebra, Geometry, Invariant Theory, Representation Theory,
and Combinatorics.
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
coset representatives.
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 symmetry to obtain the orbifold V/G which may have singularities.Geometrically, we
might replace V/G with a smooth variety,
but Hochschild
cohomology recommends an algebraic approach: replace the ring of
invariant polynomials S^G with the
natural
semi-direct product algebra S#G.
Hochschild
cohomology
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
Systems.)
My work has been supported by several
organizations:
National Science Foundation:
Research Grant
(DMS-1101177), Principal Investigator, 2011--2014
Research
Grant (DMS-0800951), Principal
Investigator,
2008--2011
Research Grant (DMS 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: 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.
TALKS GRADUATE
STUDENTS
Masters/Ph.D. Advisor for: Paisa
Seelungsawat
(BS from
MIT, MS2005
from UNT, PhD2011 from Univ.
of South Carolina)
Briana
Foster-Greenwood (BS from UNT, PhD
2012 from UNT) 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.
MATHEMATICAL
RESEARCHWorkshop
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?
PERSONAL
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
eel.
Hyperbolic
Space:
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."