Triangle Lectures in Combinatorics (TLC)

Second meeting: September 25, 2010 at Duke

Lecture room: Physics 128, Duke University

Duke Campus Map

Schedule:

9:15-10am, coffee

10-11am, Alexander Barvinok (University of Michigan), ``Maximum entropy principle in combinatorial enumeration''

11-11:30am, coffee break

11:30am-12:30pm, Anne Shiu (Duke), ``Counting positive roots of polynomials with applications for biochemical systems''

12:30-2:30pm, lunch

2:30-3:30pm, Sami Assaf (MIT), ``A kicking basis for certain Garsia-Haiman modules''

3:30-4pm, coffee break

4-5pm, Persi Diaconis (Stanford), ``A probabilistic interpretation of Macdonald polynomials''

Saturday evening, informal group dinner

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To register in advance: send email to Patricia Hersh, plhersh@ncsu.edu

Suggested hotels near Duke: Hilton Durham, 3800 Hillsborough Road; Millenium Hotel, 2800 Campus Walk Avenue; King's Daughters, 204 N. Buchanan Blvd; Brookwood Inn, 2306 Elba Street

Parking:Conference participants can park in the parking garage across from the Physics Building (PGIV) off of Science Drive. This is a visitor pay lot. You can find a map of parking lots at Duke here http://parking.duke.edu/parking/maps/map_big.pdf , PGIV is the large grey box on Science Drive. For more parking information please consult http://parking.duke.edu/.

Public Transportation: There are several buses which service the Duke campus. In particular there are free buses from East Campus (located near Downtown Durham) to West Campus (location of Physics Bldg), as well as the Bull City Connector a free bus connecting Downtown Durham to West Campus. For more information on these buses (and RDU shuttle or taxi services) please consult http://parking.duke.edu/buses_vans/index.php or email Sonja Mapes at smapes@math.duke.edu.

Pre-registered participants:

Ed Allen, Wake Forest
Sami Assaf, MIT
Avanti Athreya, Duke
Eric Bancroft, NCSU
Erin Bancroft, NCSU
Alexander Barvinok, University of Michigan
Sam Behrend, UNC Chapel Hill
Michael Benfield, NCSU
Hoda Bidkhori, NCSU
Rod Canfield, University of Georgia
Shirshendu Chatterjee, Duke
Josh Cooper, University of South Carolina
Kevin Costello, Georgia Tech
Ruth Davidson, NCSU
Persi Diaconis, Stanford
Cihan Eroglu, East Tennessee State University
Alex Fink, NCSU
Chris Fox, NCSU
Anant Godbole, East Tennessee State University
Elizabeth Harris, East Tennessee State University
Cade Herron, East Tennessee State University
Patricia Hersh, NCSU
Gabor Hetyei, UNC Charlotte
Bill Hightower, High Point University
J.T. Hird, NCSU
John Hutchens, NCSU
Badal Joshi, Duke
Chirag Lakhani, NCSU
Haiyin Li, East Tennessee State University
Martha Liendo, East Tennessee State University
Anna Little, Duke
Sonja Mapes, Duke
Sarah Mason, Wake Forest
Jonathan Mattingly, Duke
Jed Mihalisin, Meredith College
Ezra Miller, Duke
Kailash Misra, NCSU
Walter Morris, George Mason University
Michael Mossinghoff, Davidson College
Carlos Nicolas, UNC Greensboro
Rick Norwood, East Tennessee State University
Matthew O'Meara, UNC Chapel Hill
Christopher O'Neill, Duke
Daniel Orr, UNC Chapel Hill
Bob Proctor, UNC Chapel Hill
Scott Provan, UNC Chapel Hill
Nathan Reading, NCSU
Richard Rimyani, UNC Chapel Hill
Alissa Rochney, East Tennessee State University
Carla Savage, NCSU
Keith Schneider, UNC Chapel Hill
Joe Seaborn, UNC Chapel Hill
Anne Shiu, Duke
David Sivakoff, Duke
Seth Sullivant, NCSU
Prasad Tetali, Georgia Tech
Ryan Vinroot, College of William and Mary
Matt Watson, NCSU
Matt Willis, UNC Chapel Hill

Fall 2010 TLC organizing committee: Patricia Hersh (NCSU), Sonja Mapes (Duke), Ezra Miller (Duke).

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Abstracts:

Sami Assaf, ``A kicking basis for certain Garsia-Haiman modules''

In the early 90s, Garsia and Haiman constructed a family of bi-graded modules for the symmetric group indexed by partitions as a means of proving the Macdonald Positivity Conjecture. They conjectured that the dimension of the Garsia-Haiman module indexed by a partition of n is n! and showed that this would imply that the graded character is the Macdonald polynomial indexed by the same partition. The n! Conjecture was eventually proven by Haiman in 2002 using advanced techniques in algebraic geometry. In this talk, we give an elementary construction of a basis for Garsia-Haiman modules indexed by coulrophobic partitions. This basis provides an elementary proof of the dimension and, consequently, of Macdonald Positivity for these shapes. In addition, the construction gives a six term recurrence relation for the Hilbert Series of these modules and their intersections, resolving several conjectures for the Science-Fiction heuristic of Bergeron and Garsia.

Alexander Barvinok, ``Maximum entropy principle in combinatorial enumeration''

In a series of papers with J. A. Hartigan (Yale), we apply the maximum entropy principle to find asymptotic formulas for a variety of problems of combinatorial enumeration, such as finding the number of non-negative integer matrices with prescribed row and column sums or finding the number of graphs with prescribed degrees of vertices. The idea of the method is to encode the set of objects by the non-negative integer or 0-1 points in an affine subspace of Euclidean space, approximate the counting probability measure on the set by the (usually, much simpler) maximum entropy measure on the ambient space with the expectation in the subspace and estimate the cardinality of the set by using a Central Limit Theorem type argument. The method also allows us to describe the structure of a random object, such as a random non-negative integer matrix with prescribed row and column sums or a random graph with prescribed degrees.

Persi Diaconis, ``A Probabilistic Interpretation of Macdonald Polynomials''

The two parameter Macdonald polynomials are a central object of algebraic combinatorics. They are defined indirectly as the Eigen functions of a somewhat mysterious family of differential operators. Arun Ram and I have found a natural random walk on partitions, which has eigenvectors the coefficients of the Macdonald polynomials expanded in the power sums. The random walk is a version of the Swedsen-Wang algorithm of statistical physics. Using the many tools known for Macdonald polynomials gives a sharp analysis for the walk.

Anne Shiu, ``Counting positive roots of polynomials with applications for biochemical systems''

A complete root classification of a parametrized real univariate polynomial describes the number of real roots of the polynomial as a function of its coefficients. For instance, the number of real roots of a quadratic polynomial depends only on the sign of its discriminant. This talk focuses on an application of root classification for the analysis of biochemical systems. One class of such systems are the multisite phosphorylation systems, which play an important role in transmitting information in biology. We extend work of Wang and Sontag (2008) on the capacity of these systems to exhibit multiple steady states. This is joint work with Carsten Conradi, Alicia Dickenstein, and Mercedes Pérez Millán.