The American Women in
Mathematics organizers a workshop at the joint meetings each January
following up on Research Collaboration Conferences for Women.
These workshops feature
both junior and senior women speakers from one of the Research Networks
supported by the AWM ADVANCEgrant. Graduate
students may apply to present posters on their research at these
workshops each year. TORA = Texas-Oklahoma
Representations and Automorphic forms
I'm on the steering committee and
organizer for a conference
series supported by the National Science Foundation
and Oklahoma State University, University of Oklahoma, and University
of North Texas (UNT):
TORA I
meeting: September 17--18, 2011, UNT. (Plenary speakers: Nils Skoruppa, Sol
Friedberg, Martin Raum.)
TORA IV
meeting: March 23--24, 2013, UNT. (Plenary speakers: Ken Ribet, Nolan
Wallach, Susie Kimport.)
TORA
VII meeting: April 8--April 10, 2016, UNT. (Plenary speakers: Vyjayanthi Chari,
Kon Ono, Nickolas Andersen.)
MATHEMATICAL RESEARCH
I work 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 also work with 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 theory and
arrangements of hyperplanes. (Scott
Crass can explain relations with Dynamical Systems.)
My work has been supported by
Simons
Foundation:
Collaboration Grant for Mathematicians, Award Number 429539, 2016--2021 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: Research Grant, Principal Investigator, 2002--2004 Simons
Foundation:
Collaboration Grant for Mathematicians, Award Number 429539. Alexander von
Humboldt Foundation: Research Fellowship (at RWTH Aachen
University), 2009 Texas
Coordinating Board: Advanced Research Program Grant, Principal Investigator,
2008--2010.
TALKS GRADUATE
STUDENTS
Masters/Ph.D. Advisor for:
Paisa Seeluangsawat
(BS from MIT, MS 2005
from UNT, PhD 2011 from Univ.
of South Carolina)
American Mathematical Society Special Sessions recently
co-organized:
"Noncommutative and Homological Algebra" with Sarah
Witherspoon, in Denton, Texas, at University of North Texas, September
9--10, 2017.
"Noncommutative Algebra, Deformations, and Hochschild
Cohomology" with Sarah Witherspoon, in Lubbock at Texas Tech
University, April 11--13, 2014.
"Combinatorial Avenues in Representation Theory" with
Nathaniel Thiem and Richard Green, in Boulder, Colorado, April 13--14,
2013.
"Hecke Algebras and Deformations in Geometry and Topology" with Matt Douglass, in St. Paul, Minnesota, April 10-11, 2010.
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 the 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."