Sara Billey Math 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

As of September 2009, I have been promoted to (full) Professor of Mathematics at the University of Washington in the sometimes sunny city of Seattle. Here are some of my current projects

Teaching: This year I will be teaching two quarters of the three quarter combinatorics sequence for graduate students called Math 581: Foundations in Combinatorics.
Seminar: I organize the Combinatorics Seminar at UW with Isabella Novik and Michael Goff.
Editing: I am an editor for Advances in Math and for a new journal called the "Journal of Combinatorics". If you would like to submit something to either 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 smorgasbord of mathematical experiences. This year my topic will be "Sudoku" which has a lot to do with math despite what some publishers claim.
FPSAC 2010: Vic Reiner and I are co-chairs of the program committee for Formal Power Series and Algebraic Combinatorics to be held on August 2 - 6, 2010.
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.
In the news: During the fall and winter of 2009-2010 I taught Math 381: Discrete Mathematical Modeling. In order to learn about the modeling process, we 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 challenges faced by non-profit organizations, government agencies, small businesses, or the university. If you have a problem that might be suitable for this class to study, please let me know. We got some good press from the poster sessions:
- Taking Math to the Streets:Students solve real world problems with mathematics modeling in UW Week (Jan. 8, 2009)
- Students Use Math to Help Community Partners Solve Problems UW College of Arts and Sciences web page (March 17, 2009)

Research Overview

My research is in algebraic combinatorics. Combinatorics is the study of counting and bijective proofs, so an algebraic combinatorialists 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

An Introduction to Affine Grassmannians written for MSRI -- Connections Workshop, January 2009.
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 (similar 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 Algebraic 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.
Math and Juggling Summarizing the references for my Mathday talk 2009.
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.
Recently, I lost my status as the most famous Sara in Mathematics as measured by my pagerank in the google search on "sara math". So, I am attempting to reverse this trend with the following text: sara math, sara math, sara math, sara math, sara math, sara math, sara math, sara math, sara math. I do a little better in the Live Search alogorith when they don't try to correct the spelling of the name Sara. Please note, I am still the most famous Billey in Math by this measure. I hope to maintain that status for some time to come. UPDATE: Three weeks later, I am back to being the highest rated Sara in Math according to Google.