Sara Billey Math Homepage
Prof. SARA BILLEY
Department of Mathematics
Padelford C445
University of Washington
Box 354350
Seattle, WA 981954350
Phone: 2066163107
Fax: 2065430397
email: billey atsign math.washington.edu
Current Projects
I am a Professor of Mathematics at the University of Washington in the sometimes
sunny city of
Seattle. Here is my curriculum
vitae which "provides an overview of a person's life and
qualifications" according to wikipedia. See also my Google scholar page.
Below are some of my current projects:
 Teaching: This winter 2017 I will be teaching
Root Systems, Weyl Groups and Coxeter Groups .
 Seminar: I organize the Combinatorics Seminar at
UW with Isabella Novik and Hailung Zhang.

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, the
GaleShapley algorithm and
computer proof techniques. This year, I gave the plenary
lecture. Video below.
 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 nonprofit 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:
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, KazhdanLusztig 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.
Recently, I have been interested in problems which are not decidable by any
possible algorithm in finite time.
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
 Enumeration of Parabolic Double Cosets for Coxeter Groups given at Herstmonceaux Castle in England at "Algebraic Combinatorics and Group Actions Conference" June 12, 2016.
 LSU Student Colloquium Lectures: Trees, Tanglegrams, and
Tangled Chains and
Enumeration of Parabolic Double Cosets, Leap Day February 29, 2016.
 Trees, Tanglegrams, and
Tangled Chains Discrete Math Conference at Worcester Poly
Tech on September 12, 2015. See also the
video version from BIRS Workshop on "Positivity in Algebraic
Combinatorics" August 1516, 2015.
 Computer Proofs:
Coming Soon to a Theorem Near You Mathday, 2015.
A video version comes in two parts: Part I and
Part II .
 Quasisymmetric Functions
June 913, 2014. Three lecture series at ECCO at Universidad de Los Andes, Bogota Colombia.
Also here.
 Math inspired by origami
Mathday March 24, 2014. Based on work by Butler,
Demaine, Graham, and Tachi.
 CoxeterKnuth
Graphs and a Signed Little Bijection AMSMAA Joint Meetings
January 17, 2014. Joint with Zach Hamaker, Austin Roberts and
Ben Young. Extended version given at Stanford and Berkeley in
February: HERE.
 Patterns in Permutations and Diagrams ICERM
Febrary 11, 2013,
Sage Demo
 Consequences of the
LakshmibaiSandhya Theorem Four lectures plus supporting
materials written for MSJSI 2012
Schubert calculus held in Osaka, Japan. July 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 LakshmibaiSandhya 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
Also, check the
Mathematics ArXiv
 Enumeration of Double Cosets in Coxeter Groups with Matjaz
Konvalinka, T.Kyle Petersen, William Slofstra, and Bridget Tenner. Accepted to FPSAC 2016 as a poster. (More details coming soon.)
 A bijective proof of Macondald's Reduced Word Formula with Ben Young and Alexander Holroyd. Accepted to FPSAC 2016 as a talk. (More details coming soon.)

On the
enumeration of tanglegrams and tangled chains with Matjaz
Konvalinka, Frederick A Matsen IV, posted July 2015.
The Contributions of Stanley to the Fabric of
Symmetric and Quasisymmetric Functions
with Peter McNamara, posted May 2015. For a volume in honor of Richard Stanley.

Factoring Peak Polynomials
with Matthew Fahrbach and Alan Talmage. To appear Experimental Math 2015.
Additional data for Peak Polynomials and their complex zeros.

CoxeterKnuth graphs and a signed Little map for type B reduced words
with Zachary Hamaker, Austin Roberts, and Benjamin Young. Electronic Journal of Combinatorics, v. 21, issue 4 (2014), paper P4.6

Consequences of the LakshmibaiSandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
with Hiraku Abe. Preprint date, March 2014. To appear in the
Proceedings of the Mathematical Society of Japan.

On meteors, earthworms and WIMPs
with Krzysztof Burdzy, Soumik Pal, and Bruce Sagan. Annals of Probability, vol . 25, num.4. 2015.

Permutation Patterns, Stanley Symmetric
Functions and Generalized Specht Modules
with Brendan Pawlowski. Preprint. To appear in J. Combinatorial
Theory Series A. Extended abstract accepted as a
talk at FPSAC 2013. Associated pattern data.

Fingerprint Databases for Theorems
with Bridget Tenner. Notices of the AMS, Sept. 2013.

Permutations With Given Peak Set
with
Krzystof Burdzy and Bruce Sagan. J. Integer Sequences, Vol. 16
(2013).

Affine dual equivalence and kSchur functions. with
Sami Assaf. Journal of Combinatorics, special issue in
honor of Adriano Garsia. Vol 3, Num 3, p. 343399, 2012
 Pattern
characterization of rationally smooth affine Schubert
varieties of type A. with
Andrew Crites. Journal of Algebra,
Volume 361, 1 July 2012, Pages 107133.
 Singularities
of generalized Richardson varieties with
Izzet Coskun. Communications in Algebra,
Volume 40, Issue 4, April 2012, pages 14661495.
 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) 31:169216.

Permutations with KazhdanLusztig 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 KazhdanLusztig 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, 495524. 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 2154. 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. 447466.
 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, 251293.
 Lower bounds for
KazhdanLusztig polynomials from patterns
. Joint with Tom Braden. Transform. Groups vol. 8 (2003) no. 4, 321332.
 Maximal Singular Loci of
Schubert Varieties in SLn/B (with Gregory Warrington) in
Trans. AMS. 355 (2003), no. 10, 39153945.
 KazhdanLusztig Polynomials for
321hexagonavoiding permutations, (with Gregory Warrington) in J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111136.
 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. 205224, 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. 141156.

Vexillary elements of the hyperoctahedral group
Sara Billey and Tao Kai Lam. J. of Algebraic Combinatorics, v.8
(1998) pp. 139152.
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. 6984.

Schubert Polynomials for the classical groups ,
Sara Billey and Mark Haiman
Journal of AMS Volume 8, Number 2, April 1995

RCGraphs and Schubert polynomials
Nantel Bergeron and Sara Billey
Experimental Mathematics, Vol.2 (1993), No. 4.
Errata with 2 corrections.

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

Zometool Competition: I often host the annual Zometool Competition for
SIMUW;
our high school math camp.
 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
algorith 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.