In the program below, each contribution links to the corresponding extended abstract. The complete proceedings are available here (17.5 Mo). If you find a mistake, please send an email to fpsac13[at]liafa.univ-paris-diderot.fr.
| 17:00–20:00 | Welcome reception in le petit bain (registration mandatory) |
| 08:00–09:00 | Registration and breakfast |
| 09:00–10:00 |
Self-avoiding walks on the honeycomb lattice
In 2010, Duminil-Copin and Smirnov proved a long standing conjecture, according to which the number of \(n\)-step self-avoiding walks (SAWs) on the honeycomb lattice grows like \(\mu^n\), up to sub-exponential factors, where \(\mu=\sqrt{2+\sqrt 2}\).
Their proof is in fact rather simple, but also very original, at least to a combinatorialist's eyes. At the heart of the proof is a remarkable identity, that relates several generating functions of SAWs \emph{evaluated at the critical point} \(1/\mu\). We will discuss this identity and some of its extensions, with applications to SAWs interacting with a surface, and to the \(O(n)\) loop model. |
| 10:00–10:30 | Coffee break |
| 10:30–11:00 | Combinatorics of non-ambiguous trees |
| 11:00–11:30 | Fully commutative elements and lattice walks |
| 11:30–12:00 | Structure and enumeration of (3+1)-free posets |
| 12:00–14:00 | Lunch Break |
| 14:00–15:00 |
Boundaries of branching graphs
A branching graph is an infinite graded graph, sometimes with an additional structure. The boundary of such a graph describes all possible ways of escaping to infinity along "regular" monotone paths. This notion emerged about 30 years ago in the work of Vershik and Kerov on characters of the infinite symmetric group. I will survey old and new results related to boundaries of concrete graphs, and state open questions. The problems here originate from representation theory and probability theory, while the methods are mainly of combinatorial nature and rely on the theory of symmetric functions and their analogs, such as supersymmetric and quasisymmetric functions.
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| 15:00–15:30 | Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory |
| 15:30–16:00 | Coffee break |
| 16:00–16:30 | The partition algebra and the Kronecker product |
| 16:30–17:00 | The module of affine descent classes of a Weyl group |
| 17:00–19:00 |
| 09:00–10:00 |
Quiver mutation and combinatorial DT-invariants
A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers. It appeared in physics in Seiberg duality in the nineties and in mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002. We will show how, for large classes of quivers \(Q\), using quiver mutation and quantum dilogarithms, one can construct the combinatorial DT-invariant, a formal power series intrinsically associated with \(Q\). When defined, it coincides with the "total" Donaldson-Thomas invariant of \(Q\) (with a generic potential) provided by algebraic geometry (work of Joyce, Kontsevich-Soibelman, Szendroi and many others). We will illustrate combinatorial DT-invariants on many examples and discuss their links to quantum cluster algebras and to (infinite) generalized associahedra.
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| 10:00–10:30 | Coffee break |
| 10:30–11:00 | Denominator vectors and compatibility degrees in cluster algebras of finite types |
| 11:00–11:30 | Euler flag enumeration of Whitney stratified spaces |
| 11:30–12:00 | Network parameterizations for the Grassmannian |
| 12:00–14:00 | Lunch break |
| 14:00–15:00 |
Beyond \(q\): special functions on elliptic curves
An important thread in modern representation theory (and combinatorics) is that many important objects have so- called q-analogues, generalizations depending on a parameter \(q\) which reduce to more familiar objects when \(q = 1\). For instance, the Schur functions (irreducible characters of the unitary group) have \(q,t\)-analogues, namely the famous Macdonald polynomials, and similarly the Koornwinder polynomials are six-parameter \(q\)-analogues of the characters of other classical groups. It turns out that many q-analogues extend further to elliptic analogues, in which q is replaced by a point on an elliptic curve. The Macdonald/Koornwinder polynomials are no exception; I’ll describe a relatively elementary approach to those polynomials and how to modify the approach to obtain an elliptic analogue.
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| 15:00–15:30 | Moments of Askey-Wilson polynomials |
| 15:30–16:00 | Relating Edelman-Greene insertion to the Little map |
| 16:00–16:30 | Coffee break |
| 16:00–18:00 |
| 09:00–10:00 |
A unified bijective framework for planar maps
Planar maps are connected planar graphs embedded in the sphere (considered up continuous deformations). Planar maps have been actively studied in combinatorics ever since the seminal work of William Tutte in the sixties. Along the years, deep connections have been fruitfully exploited between planar maps and subjects as diverse as the combinatorics of the symmetric group, graph drawing algorithms, random matrix theory, statistical mechanics, and 2D quantum gravity.
In the last decade, following the work of Cori, Vauquelin and Schaeffer, many bijections have been discovered between classes of maps (e.g. triangulations, bipartite maps) and classes of trees. These bijections provide the ''proofs from the Book'' for the many simple-looking counting formulas discovered by Tutte and his followers. Moreover they proved to be invaluable tools in order to study the metric properties of maps, finding algorithms for maps, and solving statistical mechanics models on maps. In this talk, I will explain some of the aforementioned motivations for studying maps. I will then describe a bijective framework, developed jointly with Eric Fusy, which unifies and extends almost all the known bijections for planar maps. This framework relies on two ingredients: the existence of certain canonical orientations for planar maps, and a master bijection for oriented maps. |
| 10:00–10:30 | A generalization of the quadrangulation relation to constellations and hypermaps |
| 10:30–11:00 | Coffee break |
| 11:00–11:30 | Type A molecules are Kazhdan-Lusztig |
| 11:30–12:00 | A uniform model for Kirillov-Reshetikhin crystals |
| 12:00–12:30 | Permutation patterns, Stanley symmetric functions, and the Edelman-Greene correspondence |
| 12:30–12:45 |  Group photo |
| 12:45– | Free afternoon |
| 15:00–17:00 | Guided walking tour in Le Marais (registration mandatory) |
| 09:00–10:00 |
Razumov-Stroganov–type Correspondences in the 6-Vertex and O(1) Dense Loop Model
Razumov and Stroganov conjectured in 2001 a correspondence between the enumerations of Fully-Packed Loops (FPL) on a square domain (a version of the 6-Vertex Model), refined according to the link pattern, and the ground-state components of the Hamiltonian in the periodic XXZ Quantum Spin Chain at \(\Delta = -1/2\), a realisation of the O(1) Dense Loop Model (DLM) on a cylinder. Extensions have been considered later on. In particular, Di Francesco in 2004 suggested a one-parameter generalization: on the `DLM side', the ground state of the Hamiltonian H is replaced by the one of the Scattering Matrix, S(t) on the `FPL side', one also considers the refinement on the last row. Similar conjectures existed for two large families of domains: those with a `hidden dihedral symmetry', or with `vertical symmetry', respectively. Both the basic and extended conjectures have been proven, in the dihedral case, by L. Cantini and the speaker, while the vertical cases are open. We present the subject, its implications on Algebraic Combinatorics and Statistical Mechanics, and how the forementioned conjectures have been proven.
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| 10:00–10:30 | Coffee break |
| 10:30–11:00 | Rainbow supercharacters and poset analogue to q-binomial coefficients |
| 11:00–11:30 | The immaculate basis of the non-commutative symmetric functions |
| 11:30–12:00 | Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials |
| 12:00–14:00 | Lunch break |
| 14:00–15:00 |
Topological combinatorics of Bruhat order and total positivity
This talk will focus on the rich interplay of combinatorics, topology, and representation theory arising in the theory of total positivity and in particular in the study of the totally nonnegative part of a matrix Schubert variety. Along the way, we will survey what combinatorics of a closure poset can and what it cannot tell us about the topology of a stratified space. Braid moves on reduced and nonreduced words in the associated 0-Hecke algebra are interpreted topologically, yielding information about the possible relations among (exponentiated) Chevalley generators of a Lie group. The subword complexes introduced by Allen Knutson and Ezra Miller also play a role in this story, giving the face poset structure for the fibers of a map \(f_{(i_1,\dots ,i_d)}\) suggested in work of Lusztig where \(f_{(i_1,\dots , i_d)}\) is given by a product of exponentiated Chevalley generators. Sergey Fomin and Michael Shapiro conjectured that totally nonnegative spaces arising as images of these maps, or equivalently as the Bruhat decompositions of the totally nonnegative part of matrix Schubert varieties, together with the links of their cells, are regular CW complexes homeomorphic to balls having the intervals of Bruhat order as their closure posets. We will discuss the new combinatorics and topology which the proof of this conjecture revealed.
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| 15:00–15:30 | Double-dimers, the Ising model and the hexahedron recurrence |
| 15:30–16:00 | On Orbits of Order Ideals of Minuscule Posets |
| 16:00–16:30 | Coffee break |
| 16:00–18:00 | |
| 19:30– | Conference dinner in restaurant Le moulin vert(registration mandatory) |
| 09:00–10:00 |
Particles jumping on a cycle, a process on permutations and words
I will describe recent research regarding the so called TASEP on a cycle. It describes permutations (or
more generally words) on a cycle, where a small number may jump over a larger number. This process has been
studied both for probabilistic and algebraic combinatorics reasons. It exhibits a number of very nice structural and
enumerative properties, several of which are still unproved.
|
| 10:00–10:30 | Coffee break |
| 10:30–11:00 | Descent sets for oscillating tableaux |
| 11:00–11:30 | Spanning forests in regular planar maps |
| 11:30–12:00 | Cuts and Flows of Cell Complexes |
| 12:00–14:00 | Lunch break |
| 14:00–15:00 |
Recent Progress on the Diameter of Polyhedra and Simplicial Complexes
We review several recent results on the diameter of polytopes, polyhedra and simplicial complexes, motivated by the (now disproved, but not quite solved) Hirsch Conjecture.
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| 15:00–15:30 | A combinatorial method to find sharp lower bounds on flip distances |
| 15:30–15:45 | Coffee break |
| 15:45–16:15 | Matroids over a ring |
| 16:15–16:45 | On \(r\)-stacked triangulated manifolds |
| 16:45–17:00 | Best paper awards -- sponsored by Elsevier
|
| 17:00–18:00 | Closing goûter |
Chair: Marie Albenque
Chair: Anne Micheli
Chair: Guillaume Chapuy
