Sara Billey's homepage

Prof. SARA BILLEY

Department of Mathematics

Padelford C-445

University of Washington

Box 354350

Seattle, WA 98195-4350

Phone: 206-616-3107

Fax: 206-543-0397

email: billey at-sign math.washington.edu

Current Projects

I am an Associate Professor of Mathematics at the University of Washington in the sometimes sunny city of Seattle. Here are some of my current projects

Teaching: This fall and winter I will be teaching a course for undergrades caleed Math 381: Discrete Mathematical Modeling. In order to learn about the modeling process, we need to study real world problems that effect real people. We will seek out and solve problems related to the community around us. The course culminates in a final modeling project. Final projects can be inspired by some of the challenes faced by non-profit organizations, government agencies, small buisnesses, or the university. If you have a problem that might be suitable for this class to study, please let us know.
During winter quarter I will also be teaching Math 582: Coxeter Groups. This page will be updated before class starts.
Seminar: I organize the Combinatorics Seminar at UW with Isabella Novik and Michael Goff.
Editing: I am an editor for Advances in Math. If you would like to submit something to the journal, please use the web site. We are looking for high quality papers with wide appeal.
Zometool Competition: I host the annual Zometool Competition for SIMUW; our high school math camp.
Mathday: I speak (almost) every year at Mathday; one of the biggest mathematical events for high school students in the country where 1200 students come to campus for a smogsbord of mathematical experiences. This year my topic will be "Sudoku" which has a lot to do with math despite what some publishers claim.
AMS Special Session: Alexander Kleshchev, Stephanie van Willigenburg and I are organizing a special session on Combinatorial Representation Theory as part of the AMS 2008 Fall Western Section Meeting, to be held at the University of British Columbia, Vancouver, on October 4-5, 2008.
What is the value of a computer assisted proof? Includes details for a $500 Prize for a short computer proof of Fermat's Last Theorem.

Research Overview

My research is in algebraic combinatorics. Combinatorics is the study of counting and bijective proofs, so an algebraic combinatorialist counts algebraic objects. In particular, I am interested in Schubert polynomials, Schubert varieties, flag manifolds, Kazhdan-Lusztig polynomials, Stanley symmetric functions, Bruhat order, Weyl group and root systems of all types etc. I am a strong advocate of using computers to do math research, in particular for obtaining data for conjectures and computer verified proofs.

Books

Singular Loci of Schubert Varieties (joint with Lakshmibai) in series Progress in Mathematics, Birkhauser Boston, v. 182, 2000. NOW AVAILABLE from Amazon and a book store near you!! (No, I did not pay those people for their reviews.)

Upcoming/Recent Talks

Affine Colored Partitions and Affine Grassmannians written for AMS Special Session, January 2008.
A computational approach to Schubert varieties written for Sage Days 2007. Also includes open problems on computer verification techniques and SAGE Dreams.
Grassmannians and Other Schubert Varieties IMA Workshop on "What is Algebraic Geometry?"
Intersecting Schubert Varieties Chennai Mathematical Institute (simlar to talks given in Berkeley and UBC during the spring of 2005)
FPSAC 2005 Taormina, Italy
Schubert Varieties under a Microscope given at the AMS Summer Institute in Alebraic Geometry in Seattle.

Recent Publications, Preprints

Also, check the Mathematics ArXiv

Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h by Alex Woo, with an appendex by myself and Jonathan Weed. Preprint.
Affine partitions and affine Grassmannians with Steve Mitchell. Preprint. Computer code in lisp and Maple
Smooth and palindromic Schubert varieties in affine Grassmannians. with Steve Mitchell. Preprint.
Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory with Brant Jones to appear in the Special Issue of Annals of Combinatorics dedicated to Permutation Patterns, 2008.
Flag arrangements and triangulations of products of simplicies with Federico Ardila. Advances in Math. 214 (2007), no. 2, 495--524. 32S22 (14M15).
Intersections of Schubert varieties and other permutation array schemes with Ravi Vakil. IMA Volumes in Mathematics and its Applications Volume 146: Algorithms in Algebraic Geometry, 2007, pages 21--54. Code for experimentation is available here.
Smoothness of Schubert Varieties via Patterns in Root Systems . Joint with Alex Postnikov. Advances in Applied Math v. 34 (2005) p. 447-466.
A vector partition function for the multiplicities of sl_k(C). Joint with Etienne Rassart and Victor Guillemin. Journal of Algebra, vol. 278 (2004) no. 1, 251-293.
Lower bounds for Kazhdan-Lusztig polynomials from patterns . Joint with Tom Braden. Transform. Groups vol. 8 (2003) no. 4, 321-332.
Maximal Singular Loci of Schubert Varieties in SLn/B (with Gregory Warrington) in Trans. AMS. 355 (2003), no. 10, 3915-3945.
Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations, (with Gregory Warrington) in J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111--136.
The Parabolic map. (joint with Ken Fan and Jozsef Losonczy) J. of Alebra, vol. 214 (1999).
Kostant Polynomials and the Cohomology Ring for G/B. Duke Journal of Math, Volume 96, No. 1, pp. 205-224, 1999. This is the extended version of the announcement that appeared in the Proceedings of the National Academy of Science.
Kostant Polynomials and the Cohomology Ring for G/B Proceedings of the National Academy of Science, Jan.1997.
Pattern Avoidance and Rational Smoothness of Schubert varieties. Advances in Math, vol. 139 (1998) pp. 141--156.
Vexillary elements of the hyperoctahedral group Sara Billey and Tao Kai Lam. J. of Algebraic Combinatorics, v.8 (1998) pp. 139-152. Lisp code is available upon request for verifying the computer aided proof in this paper.
Transition Equations for Isotropic Flag Manifolds. Discrete Math., Special Issue devoted to the Conference in Taormina, Sicilly, in honor of Adriano Garsia, vol. 193 (1998) pp. 69--84.
Schubert Polynomials for the classical groups , Sara Billey and Mark Haiman Journal of AMS Volume 8, Number 2, April 1995
RC-Graphs and Schubert polynomials Nantel Bergeron and Sara Billey Experimental Mathematics, Vol.2 (1993), No. 4. Available upon request.
Some Combinatorial Properties of Schubert Polynomials , Sara Billey, Richard Stanley, and William Jockusch J. of Algebraic Comb. Vol. 2 Num. 4, 1993.
An Abstract Definition of Schubert Polynomials. University of California, San Diego, Ph.D. thesis 1994. Available upon request.

Consulting Policy

I am available as a mathematical consultant for a couple weeks per year. I analyze combinatorial problems, lecture on special topics, answer standard math questions, do literature reviews etc. The fee will be $1500 for the first day, paid up front. While I understand that this fee is high, I feel it is justified since I will have to take time away from my research and my students (who pay about $100/hour each).

Current and Former Ph.D Students

Other Fun Stuff

Women in Mathematics: This is a history of women in mathematics going back to the fifth century B.C.
Grandma My grandma is on display at the Smith-Zimmermann Museam Homepage. She is the one in the carriage.
Check out Jack Lee's web page on cool things to do in Seattle. I have visited almost every place on this list and recommend them all.