Anne  V.  Shepler
Icosahedron in Mirror

A mathematician 
is a blind man 
in a dark room 
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

  Professor of Mathematics,  
  University of North Texas    
  General Academic Building  471B  

  Mailing Address: 
  University of North Texas
  Department of Mathematics
  1155 Union Circle  #311430
  Denton, TX, 76203-5017  

  Phone:  (001) 940-565-4943
  Fax:      (001) 940-565-4805  
  E-mail:   ashepler AT

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


UNT students: put the course name and main idea on the subject line!
UNT students: is the answer to your question in the syllabus?
I don't respond to emails about private tutoring (non UNT students), web development, or coding.
Graduate students: nagging is allowed, sometimes encouraged.
Letters of recommendation: best ask 2-6 weeks before due date, more time is required if over a break.
I don't respond to emails asking about research/lab positions; 
these go to PhD students in math at UNT. 
Post-doc positions are another story.

Saying mathematics is about numbers is like saying literature is about letters.  Likewise, manipulating equations in theoretical mathematics can be about as important as manipulating grammar in literature.  Mathematics often has more in common with arts and music than other scientific fields: We imagine what could be possible.  Indeed, many people choose theoretical mathematics because it requires more imagination than any other field they have encountered. 
Math is also #1 in job satisfication according to  Undergrads: see Tom Forde's page.


The Women in Mathematical Sciences group at UNT is hosting a special talk on Fermat's Last Theorem: History, Attempts, Unsolved Issues by Professor Sue Geller from Texas A&M University joint with the mathematics department algebra seminar. This talk will be Friday, November 15th, 2019, 1--2 pm in GAB 461 and is appropriate for undergradautes, graduate students, and faculty. (I'm working on the WIMS webpage, stay tuned!)

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

I work in pure mathematics at intersections of 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

          Standard Research Grant (DMS-1101177), Principal Investigator (100% credit), 2011--2014
          Standard Research Grant (DMS-0800951), Principal Investigator
(100% credit), 2008--2011
          Standard Research Grant (DMS 0402819), Principal Investigator
(100% credit), 2004--2008
          Post-Doctoral Research Fellowship (Award 9971099), Principal Investigator
(100% credit), 1999--2002
          TORA Conference Grant (DMS-1132586), Co-Principal Investigator, 2011--2012      
          TORA Conference Grant (DMS-1302770), Co-Principal Investigator, 2013--2014
TORA Conference Grant (DMS-1600642), Co-Principal Investigator, 2013--2014
National Security Agency Research Grant, Principal Investigator (100% credit), 2002--2004
Simons Foundation
:  Collaboration Research Grant for Mathematicians,
Principal Investigator (100% credit), Award Number 429539.
Alexander von Humboldt Foundation
:  Research Fellowship (at RWTH Aachen University),
Principal Investigator (100% credit), 2009
Texas Coordinating Board
:  Advanced Research Program Grant, Principal Investigator (100% credit), 2008--2010.


ADVISING:   Masters/PhD/Post-doc Advisor for

(by and for grad students):
(undergrad research):

The American Women for Mathematics (AWM) organizes a workshop at the joint American Mathematical Society and Mathematical Association of America meetings each January, currently supported by an NSF AWM ADVANCE grant.   Sarah Witherspoon and I organized the poster session Jan 2017 and the workshop Jan 2018 on Noncommutative Algebra and Representation Theory with speakers:
Chelsea Walton, Sian Fryer, Van Nguyen, Gordana Todorov, Elizabeth Drellich, Khrystyna Serhiyenko, Ellen Kirkman,
Vyjayanthi Chari, Julia Plavnik, Natasha Rozhkovskaya, Pamela Harris, Monica Vazirani, Julia Pevtsova
Also see:

American Mathematical Society (AMS) Special Sessions recently co-organized:

ADVICE for PhD students in math


Outside of office hours and giving lectures, your math instructor is likely advising graduate students, refereeing papers, writing grant proposals, preparing manuscripts for publication, doing editorial work for journals, reviewing budgets, writing talks, skyping or phoning with collaborators, reviewing university programs, writing reports, designing new courses, grading papers, evaluating grant applications, preparing lectures and exams, organizing conferences, reviewing graduate and job applications, revising articles, writing letters of recommendation, reviewing PhD theses, proof reading manuscripts, attending department meetings, updating webpages, writing computer code, completing mathematical computations, and proving new theorems.  That is why many instructors ask you to make an appointment instead of just dropping by.

I attended the 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 divingMoray eels provide nice examples for constructing orbifolds.  Maybe not the wolf eel.  And, in case you were wondering, the Mason and Hamlin BB is 212 cm long. And, yeah, it does sound really "fat".  Especially with custom Isaac hammers (swoon).  Especially Pixies acoustic "Where Is My Mind".

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 (Univ 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."