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
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...
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...
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...
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.
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.
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.
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...
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...
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...
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.
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...
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...
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...
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.
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...
