Anne Shepler

*Anne V. Shepler*
$Photo$	A mathematician is a blind man in a dark room searching for a black cat which isn't there. ---Charles R. Darwin	He didn't live long enough to see my mathematical vindication. ---H.S.M.Coxeter on E.C. Escher	Associate Professor of Mathematics University of North Texas Office: General Academic Building 471B Mailing Address: University of North Texas Mathematics Department 1155 Union Circle #311430 Denton, Texas, USA 76203-5017 Phone: (001) 940-565-4943 Fax: (001) 940-565-4805 E-mail: ashepler $AT$ unt.edu

Why does the mirror reflect you left-right, and not up-down? Coxeter, Ludwig Wittgenstein, and E.C. Escher....

WHERE AM I? AMS Fall Western Sectional, Salt Lake City Utah, Oct 2011. Banf in May 2011.

TORA=Texas-Oklahoma Representations and Automorphic forms
I'm on the steering committee for a new conference series supported by the National Science Foundation and Oklahoma State University, University of Oklahoma, and University of North Texas.

First TORA conference: September 17--18, 2011, Univeristy of North Texas.
Plenary speakers: Nils Skoruppa, Sol Friedberg, Martin Raum.

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.

MATHEMATICAL RESEARCH

My work combines ideas in Algebra, Geometry, Invariant Theory, Representation Theory, and Combinatorics. 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. Topics include invariant theory, arrangements of hyperplanes, regular polytopes, Hecke algebras, coinvariant algebras, Coxeter groups, Shephard Groups, and Braid groups. (Scott Crass can explain relations with Dynamical Systems.) I'm particularily interested now in modular reflections groups (arising when the characteristic of the field divides the group order).

Recently, I've been working more in homological algebra, examining deformation theory, cohomology, and Graded Hecke Algebras (which include symplectic reflection algebras, Cherednik algebras, "Drinfeld Hecke algebras", etc.) with Witherspoon. Connections with geometry abound: Physicists often regard space as a Calabi-Yau manifold endowed with symmetries. We model this situation with the action of a finite group G acting linearily on V=C^n, complex space. We mod out by the symmetry and regard the orbifold V/G. The coordinate ring of V/G is the ring of invariant polynomials C[V]^G. The orbifold V/G may have singularities and so we replace V/G with a smooth variety. Hochschild cohomology tells us to replace the ring of invariant polynomials with the natural semi-direct product C[V]#G. This cohomology also governs the deformation theory of C[V]#G, from which various Hecke algebras arise.

My work has been supported by several organizations:
        National Science Foundation:
                  Research Grant (DMS-1101399), 2011--2014
                  Research Grant (DMS-0800951), 2008--2011
                  Research Grant (DMS 0402819), 2004--2008
                Post-Doctoral Research Fellowship (Award 9971099), 1999--2002
        National Security Agency: Young Investigators Research Grant, 2002--2004
        Alexander von Humboldt Foundation: Research Fellowship (at RWTH Aachen University), 2009
        Texas Coordinating Board: Advanced Research Program Grant ($43,469), 2008--2010.

PUBLICATIONS

TALKS

GRADUATE STUDENTS      Masters/Ph.D. Advisor for:
    Paisa Seelungawat
    (B.S. from MIT, M.S. from UNT, currently Ph.D. student at Univ. of South Carolina),
    Briana Foster-Greenwood
    (B.S. from UNT, current Ph.D. student at UNT.)
    Christine Uhl
    (Current Ph.D. student at UNT.)

ALGEBRA SEMINAR

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 an R.E.U. 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 (McAbee webcam, moray eels). Life goal: spot wolf eel (photo: Drew Weiner) in the wild. Dragon moray would also be nice. I religiously mulch.

Hyperbolic Space: Reflection groups and modular forms:

$Notknot$ $circle limit 3:$

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."

My boss and my other boss..