Anne  V.  Shepler
Icosahedron in Mirror
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

   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, TX, 76203-5017  

  Phone:  (001) 940-565-2155
  E-mail:   ashepler AT unt.edu

Why does the mirror reflect you left-right, and not up-down?

UPCOMING CONFERENCES:
Southwest Local Algebra Meeting SLAM 2023, at University of North Texas in Denton, March 4--5, 2023.

International Centre for Mathematical Sciences ICMS, Workshop on "Various Guises of Reflection Arrangements", March 13--17, 2023, Edinburgh, Scotland.

BEFORE YOU EMAIL ME:

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.
UNT 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.

WHY MATH?  

Mathematics often has more in common with arts and music than other scientific fields: We imagine what could be, possible universes that could exist.  Indeed, many people choose theoretical mathematics because it requires more imagination than any other field they have encountered.  Saying mathematics is about numbers is like saying literature is about letters.  Truly creative work takes flight after learning to use the grammar of mathematics: work with relationships and functions and theorems in a precise way.  Undergrads: see Mark Tomforde's page for students and math majors in particular.

PUBLICATIONS:

preprints and research papers.
 ALGEBRA SEMINAR:

includes ring theory, homological algebra, noncommutative algebra, number theory, representation theory, and combinatorics.		 WIMS

The Women in Mathematical Sciences group at UNT hosts special talks and gatherings.

MATHEMATICAL RESEARCH I work in pure mathematics at intersections of algebra, combinatorics, geometry, and representation theory.   Interests include noncommutative algebra, reflection groups, invariant theory, homological algebra, deformation theory, Hochschild cohomology, hyperplane arrangements, and codes in computer science build on isometry groups.

Physicists often regard space as a Calabi-Yau manifold endowed with symmetry. We model the local setting with a finite group G acting linearly 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.

Reflection groups 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, combinatorics, and the 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 949953 (sole PI), 2022--2027
   Collaboration Grant for Mathematicians, Award Number 429539 (sole PI), 2016--2022
National Science Foundation : 
   Standard Research Grant (DMS-1101177), Principal Investigator (sole PI), 2011--2014
   Standard Research Grant (DMS-0800951), Principal Investigator (sole PI), 2008--2011
   Standard Research Grant (DMS 0402819), Principal Investigator (sole PI), 2004--2008
   Post-Doctoral Research Fellowship (Award 9971099), Principal Investigator (sole PI), 1999--2002
   SLAM Conference Grant (DMS-2302498), Co-Principal Investigator, 2023
   TORA Conference Grant (DMS-1600642), Co-Principal Investigator, 2016--2019         
   TORA Conference Grant (DMS-1302770), Co-Principal Investigator, 2013--2014         
   TORA Conference Grant (DMS-1132586), Co-Principal Investigator, 2011--2012
National Security Agency :  Research Grant, Principal Investigator (sole PI), 2002--2004
Alexander von Humboldt Foundation :  Research Fellowship (at RWTH Aachen University), Principal Investigator (sole PI), 2009
Texas Coordinating Board :  Advanced Research Program Grant, Principal Investigator (sole PI), 2008--2010
TALKS 
ADVISING:   Masters/PhD/Post-doc Advisor for
TORA
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).
ALGEBRA SYMPOSIA
(by and for grad students): 		REUs
(undergrad research):
AWM 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

WHY MAKE AN APPT?

Outside of office hours and giving lectures, your math instructor may be advising graduate students, refereeing papers, writing grant proposals, preparing manuscripts for publication, completing editorial work for journals, reviewing budgets, writing talks, working with research 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  all in the same week. That is why many instructors ask you to make an appointment instead of just dropping by.

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 with Murad Özaydın and Andy Miller. 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.  And, in case you were wondering, Pixies acoustic "Where Is My Mind" sounds great on the Mason and Hamlin BB with Isaac hammers unless you are half Boxer, half lab.
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."