Plenary Abstracts

I will outline the combinatorial machinery behind a recent proof of the BCFW triangulation conjecture for the amplituhedron. I will mostly focus on the combinatorics of the dimer model on bipartite graphs embedded in a disk, including Kenyon–Smirnov primitives, origami crease patterns, and two punctured versions of planar bipartite graphs related by a surprising combinatorial duality.

We begin with the classical result of Karp and Sipser on maximum matchings in sparse Erdős–Rényi random graphs. Building on this foundation, we present recent progress: a central limit theorem for the matching number, a local limit theorem for the Karp–Sipser core, and the universal behaviour of the rescaled matching number of sparse random graphs. We conclude with several open problems and directions for future research.

In mathematics, the focus is often on finding interesting constructions for various mathematical problems. I will discuss how some tools can reduce the effort required to find the right questions, explore variants of a problem, test our ideas and find good constructions for various problems. We will look at different ways we can think about how these tools work, look at many examples, and figure out which of these setups is right for the problems you care about.

Macdonald polynomials are a famous family of symmetric polynomials which are connected to the Hilbert scheme, knot invariants, and the Hecke algebra. There are several combinatorial formulas for Macdonald polynomials, including the 2004 tableaux formula of Haglund–Haiman–Loehr and the 2018 multiline queue formula of Corteel–Mandelshtam–Williams. In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials called interpolation Macdonald polynomials, which are defined via vanishing conditions, and which generalize Macdonald polynomials, in the sense that their top homogeneous parts are exactly the Macdonald polynomials. In recent work with Houcine Ben Dali, we give the first combinatorial formula for interpolation Macdonald polynomials; our formula is in terms of signed multiline queues, and generalizes the multiline queue formula for Macdonald polynomials. We also give a tableaux formula. If time permits, I will discuss how these polynomials are connected to interacting particle systems.

Over the past few years there has been a series of remarkable breakthroughs in Ramsey theory. In this talk we will discuss several of these, including an exponential improvement for the diagonal Ramsey numbers, stunning new constructions for off-diagonal Ramsey numbers, and an exponential upper bound for induced Ramsey numbers.

The Generalized Lower Bound Theorem conjectured by McMullen and Walkup in the early 1970s and proved through work of Stanley, Kalai, Murai–Nevo, and Adiprasito, asserts that the g_i-numbers of any simplicial (d − 1)-sphere are nonnegative, and that for each i, the minimum g_i = 0 is achieved precisely by (i − 1)-stacked spheres. Kalai’s proof of the Lower Bound Theorem (the i = 2 case) via rigidity theory established a paradigm central to the field. A natural refinement asks for sharp lower bounds on g_i within the class S(j,d − 1) of simplicial (d − 1)-spheres whose missing faces all have dimension at most j; for most pairs (i,j) these bounds remain open, including the Charney–Davis conjecture for S(1,d − 1) — the class of flag (d − 1)-spheres.

In this talk, I will discuss recent progress on lower bound problems for spheres in S(j,d − 1). The proofs blend Stanley–Reisner ring theory, affine stress spaces, and rigidity theory. Results include new bounds on the g-numbers of spheres in certain classes S(j,d − 1) — including S(1,d − 1) and S(2,4) — as well as a classification of S(3,5)-spheres with g₃ = 1.

This is joint work with Isabella Novik.

We associate several families of univariate polynomials to any unitriangular matrix (lower triangular matrix with all diagonal entries equal to one). When the matrix is totally nonnegative, then the polynomials are proven to be real-rooted. When the matrix comes from a partially ordered set (poset), then these polynomials are known as chain polynomials and Chow polynomials. In this setting our results prove new cases of conjectures regarding real-rootedness of such polynomials. This is based on joint work with Leonardo Saud Maia Leite and Lorenzo Vecchi.