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Anne V. Shepler |
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A mathematician is a blind man in a dark room searching for a black cat which isn't there. ---Charles R. Darwin. |
Associate Professor of Mathematics
University of North Texas Office: General Academic Building 450 Mailing Address:
Phone: (001) 940-565-4943
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Why does the mirror reflect you left-right, and not up-down? Coxeter, Ludwig Wittgenstein, and E.C. Escher....
GREETINGS...
from TEXAS! Oak trees abound, no scorpions or tarantulas in sight. I'm in the math department at the University of North Texas. Before Texas, I spent two years at the University of California at Santa Cruz and a year at the University of Wisconsin at Madison. I graduated from sunny University of California at San Diego (although Doyle moved to Dartmouth: his free will).
- National Science Foundation, Post-Doctoral Research Fellowship (Award 9971099, $90,000), 1999--2002
- National Security Agency, Young Investigators Research Grant ($26,000), 2002--2004
- National Science Foundation, Research Grant (DMS 0402819, $105,000), 2004--2008
- National Science Foundation, Research Grant (DMS-0800951, $153,000), 2008--2011
- Alexander von Humboldt Foundation, Research Fellowship (at Univ. of Aachen) 2009
- Texas Coordinating Board, Advanced Research Program Grant ($43,469) 2008--2010
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 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.)
Recently, I've been working more in homological algebra. I study deformation theory, cohomology, and Graded Hecke Algebras with Witherspoon. We study Hochschild cohomology (and Lie brackets) of skew group algebras.
I've also been working on modular invariant theory with Hartmann. We investigate polynomials and forms invariant under reflection groups when the characteristic of the field may divide the group order.
Semi-invariants of finite reflection groups. pdf, dvi. Journal of Algebra 220, 314-326 (1999). Online at ScienceDirect.Abstract.
Logarithmic forms and anti-invariant forms of reflection groups, with Hiroaki Terao. In Advanced Studies in Pure Math., 27, Arrangements, Tokyo, 2000, Kinokuniya and North-Holland, Tokyo-Amsterdam. pdf, ps. Abstract.
The Sign representation for Shephard groups, with Peter Orlik and Victor Reiner. Mathematische Annalen, 322, 477-492, 2002. ps, dvi, pdf Abstract. Online at SpringerLink.
Graded Hecke algebras for complex reflection groups, with Arun Ram. Commentarii Mathematici Helvetici, 78, 308-334, 2003. dvi, ps, pdf Abstract. Online at SpringerLink.
Generalized exponents and forms. dvi, pdf Abstract. Journal of Algebraic Combinatorics, 22, 115--132, 2005.
Reflection groups and differential forms, with Julia Hartmann. To appear in Mathematical Research Letters 14 (2007) (17 pages).
Jacobians of reflection groups, with Julia Hartmann. pdf. Transactions of the American Mathematical Society, 360 (2008), 123--133.
Hochschild cohomology and graded Hecke algebras, with Sarah Witherspoon. pdf. To appear in Transactions of the American Mathematical Society. (31 pages)
- Special Session on "Arrangements and Related Topics", Co-organizer, AMS/MAA Joint Meetings, New Orleans, Jan. 2007.
- Special Session on "Invariant Theory", AMS/MAA Joint Meetings, New Orleans, Jan. 2007.
- MSRI (Berkeley), "Recent Developments in Arrangements and Configuration Spaces" honoring Sergey Yuzvinsky's 70th birthday, August, 2006.
- Texas Tech Univ., Outstanding Early Career Speaker, ``Invariant Theory in Perspective", Red Raider Symposium, Nov. 2004.
- MSRI (Berkeley), "Combinatorial Aspects of Hyperplane Arrangements", Nov. 2004.
- Texas Christian University, Colloquium, Oct 2004.
- Texas A&M, Combinatorics Seminar, April 2004.
- U.C. Santa Barbara, Automorphic Forms Workshop, March 2004.
- Microsoft Corporation and Univ. of Washington joint seminar, March 2004.
- Phoenix: AMS-MAA National, Special Session, Jan. 2004.
- U. of Goettingen, Germany, "INGO2003: Invariant Theory and its Interactions with Related Fields", March 2003.
- Louisiana State University, AMS Sectional, Special Session, March 2003.
- U. of Wisconsin, Madison, AMS Sectional, Special Session, Oct. 2002.
- University of Oregon, Colloquium, June 2002.
- University of Santa Clara, California, Colloquium, Feb. 2002.
- U. of Heidelberg, Germany, Algebra Seminar, June 2001.
- U. of Minnesota, Combinatorics Seminar, May 2001.
- U.C. Berkeley, Combinatorics Seminar, April 2001.
- Columbia U., AMS Sectional, Special Session "Arrangements of Hyperplanes", Nov. 2000.
- U. of Wisconsin, Madison, Lie Theory Seminar, Nov. 2000.
- U. of Wisconsin, Madison, Topology Seminar, Nov. 2000.
- U.C. Davis, California, Bay Area Discrete Math Day, Oct. 2000.
- U.C. Santa Cruz, California, Colloquium, Oct. 2000 and Jan. 2000.
- Valparaiso U., Indiana, Colloquium, Jan. 2000.
- Tokyo Metropolitan U., Japan, Colloquium, Nov. 1999.
- San Antonio, AMS-MAA National, Special Session "Geometry in Dynamics", Jan. 1999.
PERSONAL
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."
My Teutonic relatives....