At the crossroads of physics and mathematics : the joy of integrable combinatorics (Philippe60)

Liste des Contributions

11. Introduction

Catherine Pépin (IPhT)

24/06/2024 09:40

12. An overview of Philippe's contributions I

The organizers

24/06/2024 09:50

We will give an informal overview of Philippe's contributions, told from our own perspectives.
- Jean-Bernard Zuber : Philippe @ Saclay (and elsewhere), the early years
- Emmanuel Guitter : The Philippe60 integer sequence
- Jérémie Bouttier : Philippe and the joyful integrable combinatorics of 2D quantum gravity

13. An overview of Philippe's contributions II

The organizers

24/06/2024 11:15

We will give an informal overview of Philippe's contributions, told from our own perspectives.
- Paul Zinn-Justin : PDF, ASM, DPP and TSSCPP
- Rinat Kedem : Philippe's Paths to positivity

4. Combinatorics of 3-coloured quadrangulations

Mireille Bousquet-Mélou (CNRS, Université de Bordeaux)

24/06/2024 11:45

1. On the higher rank dimer and Ising models

Richard Kenyon (Yale University)

24/06/2024 14:30

Given a planar bipartite graph with a GL(n,R) local system, we define an associated Kasteleyn operator and show that its determinant enumerates certain objects
("n-multiwebs") generalizing the dimer model. Likewise on a nonbipartite graph with an Sp(2n) local system we show that the Pfaffian of an associated Kasteleyn-type matrix enumerates certain multiwebs generalizing Ising model...

14. Last passage percolation in a strip

Guillaume Barraquand (LPENS, CNRS/École normale supérieure)

24/06/2024 15:15

I will present a method for computing the stationary measures of integrable probabilistic systems with boundaries. We will focus on the case of a model called last passage percolation, where the stationary measure can be determined with the help of variants of the Cauchy and Littlewood summation identities for Schur symmetric functions. The method works as well for other models and their...

15. Domino tilings of generalised Aztec triangles and determinant evaluations

Christian Krattenthaler (Universität Wien)

24/06/2024 16:15

Di Francesco introduced Aztec triangles as combinatorial objects for which their domino tilings are equinumerous with certain sets of configurations of the twenty-vertex model. He conjectured a closed form product formula for the numbers of these tilings, respectively of these configurations. The formula was proved by Christoph Koutschan using Zeilberger's holonomic Ansatz and heavy...

9. Integrable dynamics on polygons and the dimer integrable system

Sanjay Ramassamy (IPhT)

24/06/2024 17:00

On the one hand, several discrete-time dynamical systems on spaces of polygons have been shown in the last twenty years to be integrable. On the other hand, Goncharov and Kenyon introduced ten years ago an integrable system associated with the dimer model on bipartite graphs on the torus. Building upon the notion of triple crossing diagram maps (introduced in recent works of Affolter, Glick,...

5. Elliptic Calogero-Moser system and Gauge Theories

Antoine Bourget (IPhT)

25/06/2024 09:30

There is a remarkable correspondence between massive vacua of certain supersymmetric gauge theories and equilibrium position of mechanical integrable systems. Here I focus on the elliptic Calogero-Moser system, which can be seen as the interactions of particles on a two-dimensional torus. Despite its simplicity, the equilibria exhibit a surprisingly rich structure, with connections with...

17. Integrable monopoles (online talk)

Charlotte Kristjansen (Niels Bohr Institute, Copenhagen University)

25/06/2024 10:15

A static monopole embedded in N=4 super Yang-Mills theory constitutes a one-dimensional defect, a ‘t Hooft line and gives rise to a defect conformal field theory. We demonstrate how quantizing around the monopole background can be implemented via the solution of beautiful and exactly solvable quantum mechanical problems.

18. The sinh-Gordon model and its excited states

Patrick Dorey (Durham University)

25/06/2024 11:15

This talk returns to the old idea that excited states in integrable quantum field theories can be found by a process of analytic continuation. By focussing on the sinh-Gordon model at small coupling, evidence for a uniform structure is found which suggests that a complete description will be possible.

19. Branes and Supersymmetric Gauge Theories

Amihay Hanany (Imperial College)

25/06/2024 12:00

This talk will go over various brane constructions which arise in superstring theories and give rise to quiver gauge theories, emphasizing the combinatorial aspect of these physical systems. We will try to cover connections with mathematical topics in tropical geometry, brane tilings, cluster algebras, and symplectic singularities.

6. The magic number conjecture for the m=2 amplituhedron and Parke-Taylor identities

Lauren Williams (Harvard University)

25/06/2024 14:30

The magic number conjecture says that the cardinality of a tiling of the amplituhedron An,k,m is the number of plane partitions which fit inside a k by (n-k-m) by m/2 box.
(This is a generalization of the fact that triangulations of even-dimensional cyclic polytopes have the same size.) I'll explain how we prove the magic number conjecture for the m=2 amplituhedron; we also show that all...

20. Many new conjectures on Fully-Packed Loop configurations

Andrea Sportiello (LIPN, CNRS/Université Sorbonne Paris Nord)

25/06/2024 15:15

We deal with one of the favourite objects of Philippe: Fully-Packed Loop configurations, in domains where the Razumov--Stroganov conjecture holds. Recall that the RS conjecture relates FPL's and the steady state of the O(1) dense loop model. In short, it states that the refined enumeration of FPL's according to the (black) link pattern is proportional to the aforementioned steady state. The...

21. Arctic curve fluctuations in the square-ice model

Filippo Colomo (INFN, Florence)

25/06/2024 16:15

We study the emptiness formation probability (EFP) in the six-vertex model with domain wall boundary conditions. At the ice point, i.e., when all the Boltzmann weights are equal, we are able to build an explicit, although still conjectural, expression for the EFP as the Fredholm determinant of some linear integral operator. As the geometric parameters of the EFP are tuned to the vicinity of...

10. Mating of discrete trees and walks in the quarter plane

Philippe Biane (LIGM, CNRS/Université Gustave Eiffel)

25/06/2024 17:00

I will exhibit a simple construction of planar maps using walks in the quarter plane. This allows to recover in a unified way several known bijections between walks and planar maps (possibly decorated by some combinatorial data) and also to find new bijections.

3. Random point processes in the plane and applications to birds of prey

Gernot Akemann (Bielefeld University & University of Bristol)

26/06/2024 09:30

Random point processes including determinantal ones are popular models in ecology. In this talk I will put the two-dimensional Coulomb gas at general inverse temperature $\beta\geq0$ in a such a perspective. Away from the integrable point beta=2, corresponding to the Ginibre ensemble of random matrices with complex normal entries, the Poisson point process at beta=0, very little is known...

23. Incidence geometry and tiled surfaces

Sergey Fomin (University of Michigan)

26/06/2024 10:00

We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a triangulation of a closed oriented surface, or a tiling of such a surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and...

8. Playing with free probability in noisy many-body quantum systems

Denis Bernard (LPENS, CNRS/École normale supérieure)

26/06/2024 11:15

16. Semiclassical limit of the spin Calogero-Moser system as a hybrid integrable system (online talk)

Nicolai Reshetikhin (YMSC Tsinghua University)

26/06/2024 12:00

The talk will start with a brief outline of hybrid integrable systems. Such systems
consist of an integrable classical background and a quantum "bundle" over the
phase space of this classical system. The quantum dynamics of such a system is
"driven" by the classical integrable dynamics. An example of such a system
appears in the semiclassical limit of the spin Calogero-Moser system. We...

24. Non-unitary fermions and extended symmetry

Didina Serban (IPhT)

26/06/2024 14:30

I will present a long-range integrable model based on the Temperley-Lieb algebra at the free fermionic point. In spite of the lack of translational invariance, the model possesses an extended symmetry and a very simple spectrum.

25. Ruijsenaars wavefunctions as modular group matrix coefficients

Alexander Shapiro (University of Edinburgh)

26/06/2024 15:00

The phase space of the Ruijsenaars integrable system can be identified with (a Poisson reduction of) the moduli space of $GL_n$ local systems on a punctured torus. The latter admits a structure of a cluster Poisson variety. On the algebraic level, this leads to an injective homomorphism from a spherical subalgebra of the double affine Hecke algebra into the quantized algebra of global...

7. Non-invertible symmetries in loop models

Hubert Saleur (IPhT)

26/06/2024 16:00

I will revisit some old results of Philippe, Jean-Bernard Zuber and I about degeneracies in O(n) CFTs, and interpret them using ideas of non-invertible symmetries as well as tools from the bootstrap.

26. Conclusion

26/06/2024 16:30

22. Cancelled

Andrei Okounkov (Columbia University)