All talks will be held in the Science Lecture Hall. The poster sessions will be held on Tuesday July 18 and Thursday July 20 from 5pm until 6:30pm in the Multipurpose Room at the Student Community Center. A reception will take place from 5pm until 6:30pm on Sunday July 16, 2023 in Wall Hall Main Lodge.
All recordings are posted to youtube on our playlist.
Abstracts of plenary talks
Karim Adiprasito: Parseval identitities, volume polynomials and anisotropy
The volume polynomial of a standard Gorenstein ring parametrizes the fundamental class. In terms of the torus action, it is a rational function. I will explore some identitites for this polynomial and discuss applications to combinatorics, in particular to lattice polytopes and h* polynomials.
Jonah Blasiak: Catalania
Many well-known formulas in symmetric function theory such as those for Hall-Littlewood polynomials and the Weyl character formula involve a product over all positive roots. Replacing this product with one over an upper order ideal of positive roots (of which there are Catalan many) yields new families of polynomials. We will see how this idea leads to elegant formulas for $k$-Schur functions, their $K$-theoretic versions, $\nabla s_\lambda$, and Macdonald polynomials, and explore how such formulas can pave the way to positive combinatorics.
Melody Chan: Graph complexes and moduli spaces in tropical geometry
Kontsevich’s graph complexes have a combinatorial definition that is easy to explain: they are certain rational chain complexes, generated by graphs with orientations, with differential given by a sum of 1-edge contractions. Though elementary to define, graph complexes have several different deep connections to geometry. One of these connections, discovered in joint work with Galatius and Payne, is to moduli spaces in algebraic and tropical geometry. I hope to give a broad and accessible talk on these ideas.
Eric Fusy: Enumeration of rectangulations and corner polyhedra
I will present some results on the exact and asymptotic enumeration of rectangulations (tilings of a rectangle by rectangles) and corner polyhedra (a topological version of plane partitions). These objects can be encoded as certain decorated planar maps, and I will explain how these can be reduced to models of (decorated) plane bipolar orientations, which then yields an encoding by certain quadrant walks, thanks to a bijection due to Kenyon, Miller, Sheffield and Wilson. The considered counting sequences have the nice feature that the exponential growth rate is a simple rational number (even if for some of them the generating function is not D-finite).
Joint work with Erkan Narmanli and Gilles Schaeffer
Rei Inoue: Cluster realization of Weyl group and its applications to representation theory
The cluster algebra is a commutative algebra introduced by Fomin and Zelevinsky around 2000. The characteristic operation in the algebra called ‘mutation’ is related to various notions in mathematics and mathematical physics. In this talk we introduce a realization of Weyl groups in terms of cluster mutations, for a finite dimensional semisimple Lie algebra. We briefly explain its applications to the higher Teichmuller theory introduced by Fock and Goncharov in 2003, and to the $q$-characters of quantum non-twisted affine algebras introduced by Frenkel and Reshetikhin in 1998. This talk is based on joint works with Thomas Lam, Pavlo Pylyavskyy, Tsukasa Ishibashi, Hironori Oya, and Takao Yamazaki.
Mohamed Omar: Using slice-rank and partition-rank
Recent breakthroughs in combinatorics, especially on bounds of sizes of sets avoiding particular configurations, have been afforded by the slice-rank and partition-rank methods. In this talk we introduce these concepts and the challenges that arise when using them, in hopes that audience members have access to a new tool they may find useful in their own work. Furthermore we discuss the work of the speaker in integrating partition lattices into the theory.
Anna Weigandt: Derivatives and Schubert Calculus
Schubert Calculus has its origins in enumerative questions asked by the geometers of the 19th century. Algebraic reformulations of these problems have led to a vast theory which studies symmetric polynomials and related tableau combinatorics. In this talk, we will discuss how to use derivatives to shed light on algebraic and combinatorial properties of families of polynomials.
Paul Zinn-Justin: Schubert puzzles as exactly solvable models
Schubert polynomials are a remarkable basis of the space of polynomials in countably many variables. Expanding the product of two Schubert polynomials in this basis leads to a generalisation of Littlewood-Richardson coefficients for which a combinatorial formula is desirable. We review recent progress in using methods from exactly solvable models of statistical mechanics and the related representation theory to construct “Schubert puzzles” that provide such formulae in an ever expanding range of particular cases. This is joint work with Allen Knutson.