Twenty-fifth meeting: Saturday, September 21, 2024
Location: 105 Cameron Hall, UNC Wilmington
Parking: Visitor Parking at 4941 Riegel Road, Wilmington, NC (~5-10 minute walk to Cameron)
Speakers:
Sarah Mason (Wake Forest)
Nick Mayers (NCSU)
Corrine Yap (Georgia Tech)
Shira Zerbib (Iowa State University)
Schedule (all times in EST)
09:00 - 10:00am, welcome
10:00 - 11:00am, Corrine Yap
11:00 - 11:30am, coffee break
11:30 - 12:30pm, Nick Mayers
12:30 - 2:30pm, lunch break
02:30 - 3:30pm, Shira Zerbib
03:30 - 4:00pm, coffee break
04:00 - 5:00pm, Sarah Mason
Registration and Funding: Please fill out the Google Form to register for the conference. There are questions regarding funding in the same form.
Lunch break: Shore or Wagoner dining halls, fast food at Fisher University Union. Slightly further away, there are many options along College Rd (~15-20 minute walk).
Organizing Committee: Laura Colmenarejo (NCSU), Sean English (UNC Wilmington) and Clifford Smyth (UNC Greensboro)
Other Local Organizers: Lilit Martirosyan and Erik Slivken
Funding provided by NSF
Titles and Abstracts
Sarah Mason: The quest for quasisymmetric Macdonald polynomials
Macdonald polynomials are a two-parameter generalization of symmetric functions with connections to representation theory, geometry, and combinatorics. A recent breakthrough by Corteel, Mandelshtam, and Williams connecting Macdonald polynomials to the asymmetric simple exclusion process (ASEP) has paved the way for new developments in the theory of Macdonald polynomials. In this talk, we build on these connections to introduce a quasisymmetric refinement of Macdonald polynomials as well as several new and more compact combinatorial formulas for Macdonald polynomials. This is joint work with Corteel, Haglund, Mandelshtam, and Williams.
Nick Mayers: The quantum k-Bruhat order
Finding combinatorial interpretations for the structure constants of Schubert polynomials is a long-standing open problem in algebraic combinatorics. In the case where one of the Schubert polynomials is a Schur polynomial, the structure constants are encoded in a poset called the “k-Bruhat order”. In studying the k-Bruhat order, Bergeron and Sottile were led to introduce a monoid which encodes the chain structure of the k-Bruhat order. Using the monoid structure, the authors were able to establish properties and descriptions of certain structure constants. In this talk, after outlining the developments discussed above, we discuss ongoing work concerning an analogous story for quantum Schubert polynomials and an associated quantum k-Bruhat order. This is joint work with Laura Colmenarejo.
Corrine Yap: Dynamical Thresholds for the Fixed-Magnetization Ising Model
Spin models on graphs are a source of many interesting questions in statistical physics, algorithms, and combinatorics. The Ising model is a classical example—first introduced as a model of magnetization, it can combinatorially be described as a weighted probability distribution on 2-vertex-colorings of a graph. We’ll consider a fixed-magnetization version of the Ising model—akin to fixing the number of, say, blue vertices in every coloring—and a natural Markov chain sampling algorithm called the Kawasaki dynamics. We show some surprising results regarding the existence and location of a fast/slow mixing threshold for these dynamics. Our proofs require a combination of Markov chain tools, such as path coupling and spectral independence, with combinatorial tools, such as random graph analysis and second moment methods. Joint work with Aiya Kuchukova, Marcus Pappik, and Will Perkins.
Shira Zerbib: Improved Tverberg theorems for certain families of polytopes
The celebrated Tverberg theorem states that any set of (d+1)(r-1)+1 points in R^d has a partition into r subsets whose convex hulls have a point in common. A theorem of Grünbaum, which states that every m-polytope is a refinement of an m-simplex, implies the following generalization of Tverberg's theorem: if f is a linear function from an m-dimensional polytope P to R^d and m <= (d + 1)(r - 1), then there are r pairwise disjoint faces of P whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map f is continuous and r is a prime power. In this talk we show that for certain families of polytopes the lower bound on the dimension m of the polytope can be significantly improved, both in the affine and topological cases. To this end we also prove a generalization of the topological Borsuk-Ulam for r points. Joint work with Pablo Soberón.