The links between combinatorics and statistical physics are of ancient origin (see, e.g., Rota's essay in "The mathematical sciences: A collection of essays", M.I.T. Press, Cambridge, Mass.-London, 1969, x+271 pp.). In the last few years, more and more connections have been found between the branch of combinatorics known as "algebraic combinatorics" (particularly enumeration, symmetric functions, cluster algebras, geometric combinatorics, and combinatorial representation theory) and various other areas in mathematical physics such as, for example, quantum field theory, tensor models, integrable Hamiltonian systems, conformal field theory, the KP-equation, the thermodynamic Bethe ansatz, super Yang-Mills scattering amplitudes, and the Toda lattice. These connections between combinatorics and mathematical physics are now so recognized that a name has been given to this area of interaction, and books have been written about it (see, e.g., A. Tanasa, "Combinatorial Physics", Oxford University Press, 2021, or "Physical combinatorics", Birkhäuser, Boston, MA, 2000, x+317 pp.). The aim of this online workshop is to explore some of these recent connections with the help of four of the leading experts in the area, and to encourage combinatorialists to look at mathematical physics as a rich source of interesting and challenging combinatorial problems, and mathematical physicists to look to combinatorics as a useful tool in their research.

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The Potts model and Lorentzian polynomials on cones

Petter Brändén (KTH, Stockholm)

The Potts model (or the random cluster model) for graphs and matroids has been much studied in statistical physics and combinatorics. It is believed that this model exhibits strong negative dependence properties. In this talk we will use Lorentzian polynomials to try to answer such questions. In joint work with June Huh, we proved that for q between zero and one, the partition function of the q-state Potts model is ultra log-concave. This settled a conjecture of Mason from 1972. Together with Jonathan Leake, we extend the theory of Lorentzian polynomials to convex cones. This is used to give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid, and reprove the Hodge-Riemann relations of degree one for the Chow ring of a matroid (first proved by Adiprasito, Huh and Katz).

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Positive geometries and scattering amplitudes

Thomas Lam (University of Michigan, Ann Arbor)

Scattering amplitudes are analytic functions of space-time momenta (and other data) that compute probabilities of various outcomes in elementary particle interactions. For certain quantum field theories, the tree-level scattering amplitude is a rational function that is essentially determined by various symmetries and factorization properties. Motivated by this, we defined (in joint work with Arkani-Hamed and Bai) positive geometries, which are certain semialgebraic varieties equipped with a meromorphic top-form called the canonical form, from which scattering amplitudes can be recovered. Examples of positive geometries include polytopes, positive parts of toric varieties, totally nonnegative Grassmannians, and conjecturally, amplituhedra.

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Many new conjectures on Fully-Packed Loop configurations

Andrea Sportiello (CNRS and LIPN, Université Paris Nord)

The Razumov-Stroganov conjecture revolves around Fully-Packed Loop configurations (FPL) and the steady state of the dense O(1) loop model (O(1)DLM). In short, it states that the refined enumeration of FPL's according to the (black) link pattern is proportional to the aforementioned steady state. It exists in two main flavours: "dihedral" (ASM, HTASM, QTASM on one side, and the O(1)DLM on the cylinder on the other side), and "vertical" (VSASM, UASM, UUASM, OSASM, OOASM on one side, and the O(1)DLM on the strip on the other side). Together with L. Cantini, we gave two proofs (in 2010 and 2012) of the conjecture in the dihedral cases, but, despite the efforts of ourselves and others, the vertical case is still unsolved. We recently looked back at the FPL configurations pertinent to one of the unsolved cases, namely the UASM (ASM on a 2n x n rectangle with U-turn boundary conditions on one long side), and we had the idea of looking at the refinement according to the black and white link patterns, and the overall number of loops. This doesn't seem to help in understanding the Razumov-Stroganov conjecture, but leads to many more conjectures, suggesting the existence of a remarkable generalisation of Littlewood-Richardson coefficients, somewhat in the same spirit, but apparently by a completely different mechanism, to "FPL in a triangle" studied by P. Zinn-Justin, and by Ph. Nadeau. Work in collaboration with L. Cantini.

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Combinatorics of the amplituhedron

Lauren Williams (Harvard University, Cambridge)

The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka in order to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes objects from combinatorics including cyclic polytopes and the (bounded complex of the) cyclic hyperplane arrangement. I'll give an introduction to the amplituhedron, including connections to the hypersimplex and cluster algebras.

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