Combinatorics and interactions

Introductory school at CIRM: January 9—13 2017

Organizers: Jérémie Bouttier, Guillaume Chapuy, Enrica Duchi.

--> New! The videos of lectures and researchs talks are available, see the link in the right-menu.

The school will be held in CIRM near Marseille. It will feature four courses by the following lecturers:

• Nicolas Curien,
• Piotr Śniady,
• Béatrice de Tilière,
• Dmitri Zvonkine.

Each course will consist of three 90 minute lectures and one exercise session. See below on this page for titles and summaries of the courses.

In addition to the courses, the school will feature six research talks (see abstracts below) by

• Mathilde Bouvel,
• Élise Goujard,
• Danilo Lewanski,
• Alexander Moll,
• Jonathan Novak,
• Gourab Ray.

Click here for the group photo!

Program of the week

• The timetable of the week in PDF.
• The whole program (timetable and abstracts) in PDF.
• The timetable in ICS format to import in your favourite calendar app: timetable.ics.

The abstracts of the courses and research talks are also available below on this page.

Registration

Registration is now closed. Accepted participants have been notified. The deadline to pre-register is October 31, 2016. Due to space limitation, there may be a selection of participants.

Thanks to financial support from CIRM and IHP, the school will cover the stay (accommodation and meals) for most accepted participants. As a general rule, the travel expenses will not be covered.

Outlines of the courses

• Nicolas Curien: Peeling Random Planar Maps.
  Outline:
  The spatial Markov property of random planar maps is one of the most important properties of these random lattices. Roughly speaking, this property says that, after a region of the map has been explored, the law of the remaining part only depends on the perimeter of the discovered region. The spatial Markov property was first used in the physics literature, without a precise justification: Watabiki introduced the so-called “peeling process”, which is a growth process discovering the random lattice step by step and used it to derived the so-called “two-point” function of 2D quantum gravity. A rigorous version of the peeling process and its Markovian properties was given by Angel in the case of the Uniform Infinite Planar Triangulation (UIPT), which had been defined by Angel and Schramm as the local limit of uniformly distributed plane triangulations with a fixed size. The peeling process has been used since to derive information about the metric properties of the UIPT, about percolation and simple random walk on the UIPT and its generalizations, and more recently about the conformal structure of random planar maps. It also plays a crucial role in the construction of “hyperbolic” random triangulations. In this course we review and extend these results via the new and more universal peeling process recently introduced by Budd which enables us to treat all the Boltzmann map models at once.
  
  
  
• Piotr Śniady: Characters, maps, free cumulants.
  Outline:
  - Normalized characters of the symmetric groups,
  - Kerov polynomials and Kerov positivity conjecture,
  - Stanley character polynomials and multirectangular coordinates of Young diagrams,
  - Stanley character formula and maps,
  - Jack characters - characterization, partial results.

  Lecture notes and Exercise sheet on Piotr Śniady's website.

  
  
• Béatrice de Tilière: Dimers and related models in statistical mechanics.
  Outline:

  The dimer model represents the adsorption of diatomic molecules on the surface of a crystal. Our first goal for these lectures is to present the founding result of Kasteleyn, Temperley & Fisher proving an exact formula for the partition function, the local probability formula due to Kenyon, and the discrete surface interpretation due to Thurston. Then, we will give a brief overview of the phase diagram obtained by Kenyon, Okounkov and Sheffield in the bipartite case.

  The 2-dimensional Ising model is a well known model for ferromagnetism. Our second goal is to explain the notion of Z-invariance for this model, a concept extensively developed by Baxter. To this purpose we will introduce isoradial graphs and elliptic functions.

  One way of studying the Ising model is to use Fisher's correspondence relating it to the dimer model on a decorated graph. Using this approach, our third goal is to present results obtained with C. Boutillier and K. Raschel, proving local formulas for the free energy and probabilities in the Z-invariant case.

  Exercise sheet (pdf).

  
  
• Dimitri Zvonkine: Hurwitz numbers.
  Outline:
  LECTURE 1. HURWITZ NUMBERS AND INTEGRABLE HIERARCHIES
  

  In this lecture we define Hurwitz numbers, write out its generating series using the irreducible representation of the symmetric group, give an introduction to the classical integrable hierarchies (specifically, the Korteweg - de Vries, Kadomtsev-Petviashvili, Gelfand-Dickey, and Hirota hierarchies), and prove that the generating series of Hurwitz numbers is a solution of the Kadomtsev-Petviashvili hierarchy.

  LECTURE 2. HURWITZ NUMBERS AND THE INTERSECTION THEORY ON MODULI SPACES OF CURVES
  

  In this lecture we will introduce moduli spaces of curves, the Deligne-Mumford compactification, and some natural cohomology classes on these spaces. We will present the ELSV formula that expresses Hurwitz numbers as an integral of cohomology classes over the moduli space of curves.

  LECTURE 3. HURWITZ NUMBERS AND THE TOPOLOGICAL RECURSION
  

  The topological recursion is a way to construct invariants starting from a plane curve. The construction originates from the study of matrix models, whose free energy satisfies the so-called "loop equation" that can be written out exclusively in terms of the spectral curve of the matrix model. This allows one to generalize the loop equation to any plane curve and construct its solution. We will present a proof of the Bouchard-Mariño conjecture stating that Hurwitz numbers appear as a solution of the loop equation for a particular plane curve.

  Exercise sheet (pdf).

  
  

Outlines of the research talks

• Mathilde Bouvel (Universität Zürich) Studying permutation classes using the substitution decomposition
  Outline:
  The notion of “pattern” in a permutation provides a natural notion of substructure for permutations. Permutation classes are downsets for the corresponding partial order. Per- mutation patterns and permutation classes were first defined in the seventies, in connection with sorting devices. But since then, most of the work done in the area is in enumerative combinatorics (although probabilists recently started to be interested in the topic). In my talk, I will present the substitution decomposition of permutations, which allows to encode permutations as trees. I will show several examples of results on permutation classes that can be derived using substitution decomposition. These will include exact enumerative results for specific permutation classes, general enumerative results for families of permu- tation classes, and a probabilistic result describing the “limit shape” of permutations taken in the class of so-called separable permutations.
  
  
  
• Élise Goujard (Université Paris-Sud Orsay) Flat surfaces and combinatorics.
  Outline:
  Billiards in polygons are related to dynamics of the linear flow on flat surfaces. Through some examples of counting problems on flat surfaces and on moduli spaces of flat surfaces, we will see how combinatorics can lead to interesting dynamical results in this setting.
  
  
  
• Danilo Lewanski (Universiteit van Amsterdam) Orbifold Hurwitz numbers, topological recursion and ELSV-type formulae.
  Outline:
  ELSV-type formulae express Hurwitz numbers of certain type in terms of the intersection theory on moduli spaces of curves. ELSV-type formulae and Topological Recursion for the appropriate spectral curve are in many examples proved to be equivalent results. In particular, a specialisation of the ELSV- type formula due to Johnson, Pandharipande and Tseng (JPT) can be shown to be equivalent to the topological recursion for the r-orbifold Hurwitz numbers, using Chiodo classes. We will discuss this equivalence and how to prove the topological recursion, obtaining a new proof of the specialised JPT formula. Based on two joined works with (subsets of) P. Dunin-Barkowski, A. Popolitov, S. Shadrin, D. Zvonkine.
  
  
  
• Alexander Moll (Institut des Hautes Études Scientifiques) A new spectral theory for Schur polynomials and applications.
  Outline:
  After Fourier series, the quantum Hopf-Burgers equation v_t + v v_x = 0 with periodic boundary conditions is equivalent to a system of coupled quantum harmonic oscillators, which may be prepared in Glauber's coherent states as initial conditions. Sending the displacement of each oscillator to infinity at the same rate, we (1) confirm and (2) determine corrections to the quantum-classical *correspondence principle*. After diagonalizing the Hamiltonian with Schur polynomials, this is equivalent to proving (1) the concentration of profiles of Young diagrams around a limit shape and (2) their global Gaussian fluctuations for Schur measures with symbol v:T-> R on the unit circle T. We identify the emergent objects with the *push-forward* along v of (1) the uniform measure on T and (2) H^{1/2} noise on T. Our proofs exploit the integrability of the model as described by Nazarov-Sklyanin (2013). As time permits, we discuss structural connections to the theory of the topological recursion.
  
  
  
• Jonathan Novak (University of California, San Diego) Monotone Hurwitz numbers and the HCIZ integral.
  Outline:
  The Harish-Chandra/Itzykson-Zuber integral is a basic special function in representation theory and random matrix theory. In representation theory, it appears in the Harish- Chandra/Kirillov character formula for the general linear groups; in random matrix theory, it describes the spectrum of coupled random matrices with the so-called “AB interaction.” I will describe joint work with I. Goulden and M. Guay-Paquet which reveals that the HCIZ integral is a generating function for a desymmetrized version of the double Hurwitz numbers, known as monotone double Hurwitz numbers. This combinatorial model for the HCIZ integral is especially useful for analyzing its asymptotic behaviour in various scaling limits.
  
  
  
• Gourab Ray (University of Cambridge) Universality of fluctuations of the dimer model.
  Outline:
  We introduce a new robust technique of tackling the problem of universality of fluctuations of the height function of dimer models on general graphs (on the hexagonal lattice, this model is also known as lozenge tilings). This will use the exact solvability of the model through various forms of bijections and not through the popular method of analysing Kasteleyn matrices. For this, we exploit the new GFF/SLE coupling results of imaginary geometry due to Dubedat, Miller and Sheffield. As an application, I will try to convince you of the universal behaviour of the fluctuations under only a CLT assumption on a related graph (called T graphs) for the Gibbs measure on lozenge tiling with any mean slope. Time permitting, I will also discuss an ongoing work on the universality of height function fluctuation on a surface with more general topology (e.g. a torus). This is joint work with Nathanael Berestycki and Benoit Laslier.
  
  
  

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