Prof. SARA BILLEY

Current Projects

• Teaching: This fall 2012 I will be teaching Root System and Coxeter Groups.

• Seminar: I organize the Combinatorics Seminar at UW with Isabella Novik, and Brendan Pawlowski.

• 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 click on the journal name. 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. In the past I have talked about mathematical card tricks, sudoku, math of the web, and computer proof techniques.

• ICERM 2013: I am co-organizing a semester at the newest NSF funded math institute on "Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series" (January 28, 2013 - May 3, 2013)

• 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: I often teach 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

• Patterns in Permutations and Diagrams ICERM Febrary 11, 2013, Sage Demo
• Consequences of the Lakshmibai-Sandhya Theorem Four lectures plus supporting materials written for MSJ-SI 2012 Schubert calculus held in Osaka, Japan. July 2012.
• Computer Proofs vs Human Proofs; Who will win? written for MathDay 2012.
• Recent advances in symmetric functions and tableaux combinatorics Current Problems Seminar, UW. March 1, 2012.
• An Introduction to the Combinatorics and Geometry of Schubert Varieties Nebraska Conference for Undergraduate Women in Mathematics, January 28, 2012.
• Consequences of the Lakshmibai-Sandhya Theorem AWM 40th Anniversary Conference --Sept 18, 2011.
• Mathematics of the Web written for Mathday -- March 21, 2011.
• Rank Varieties written for CMS Special Session in Vancouver, December 2010.
• A Stratification of the Space of Branched Polymers written for Combinatorics 2010: Advances, Trends and Speculations -- March 26, 2010.
• 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

• Fingerprint Databases for Theorems with Bridget Tenner. To appear in Notices of the AMS.
• Permutations With Given Peak Set with Krzystof Burdzy and Bruce Sagan. Preprint.
• Affine dual equivalence and k-Schur functions. with Sami Assaf. Journal of Combinatorics, special issue in honor of Adriano Garsia. Vol 3, Num 3, p. 343-399, 2012
• Pattern characterization of rationally smooth affine Schubert varieties of type A. with Andrew Crites. Journal of Algebra, Volume 361, 1 July 2012, Pages 107-133.
• Singularities of generalized Richardson varieties with Izzet Coskun. Communications in Algebra, Volume 40, Issue 4, April 2012, pages 1466-1495.
• REU Report on Branched Polymers with Tom Boothby, Chris Fox and Morgan Eichwald. Manuscript. See also Polymer code in SAGE
• Smooth and palindromic Schubert varieties in affine Grassmannians. with Steve Mitchell. Journal of Algebraic Combinatorics (2010) 31:169-216.
• Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h by Alex Woo, with an appendex by myself and Jonathan Weed. The Electronic Journal of Combinatorics Volume 16(2), 2009. Published May 12, 2009.
• Affine partitions and affine Grassmannians with Steve Mitchell. Electronic Journal of Combinatorics, Special Issue in honor of Anders Bjorner's 60th Birthday, R18, Volume 16(2), 2009. Computer code in lisp and Maple
• 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

Current and Former Ph.D Students

• Greg Warrington.
• Etienne Rassart.
• Brant Jones.
• Beth Kelly.
• Kurt Luoto.
• Andrew Crites.
• Brendan Pawlowski.
• Austin Roberts.

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 Bing 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.
• Music: I sing and play the flute with the Intergenerational Choir and the Flute Group at the University Unitarian Church. I have posted some of our performances here.
• Mathbabe : My friend Cathy O'Neil has a really amazing way with words and a very interesting set of experiences as a mathematician. I hope she writes a book one day. Until then, check out her blog.
• Advice: I find that I collect advice like some people collect butterflies, as a hobby. I also like to share my advice collection with other. So, I am putting some of it into a web page.