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
 
Department of Mathematics
  1155 Union Circle  #311430
  Denton, Texas, USA 76203-5017 

   Phone:    (001) 940-565-4943
   Fax:        (001) 940-565-4805  

   E-mail: asheplerAT unt.edu

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


ALGEBRA SEMINAR

PUBLICATIONS
WINART
Women in Noncommutative Algebra and Representation Theory workshop.
BIRS (Banff International Research Station), Alberta, Canada, March 27-- April 1, 2016.

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): 


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 theory and 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:  Research Grant, Principal Investigator, 2002--2004
Alexander von Humboldt Foundation:  Research Fellowship (at RWTH Aachen University), 2009
Texas Coordinating Board
:  Advanced Research Program Grant,
PI, 2008--2010.

TALKS 

GRADUATE STUDENTS      Masters/Ph.D. Advisor for:

ALGEBRA SYMPOSIA
(by and for grad students):
REUs
(undergrad research):


American Mathematical Society Special Sessions recently co-organized:

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 the wolf eel.  

Hyperbolic Space:              Reflection groups and modular forms:

Notknotcircle 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."


  Meter