Duke Campus Map

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

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

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.