# Talks

Titles, abstracts and slides (if available):

## George Andrews: *Ramanujan's Lost Notebook in Five Volumes - Thoughts and Comments.*

Abstract:Bruce Berndt and I have recently completed the fifth and final volume on Ramanujan's Lost Notebook. All of Ramanujan's assertions (with perhaps one or two exceptions) have been proved or, in very rare instances, refuted or corrected. Among these hundreds of formulas there are a number that stand out. For example, the recent explosion of results on mock theta functions and mock modular forms has it origin in the Lost Notebook. The "sums-of-tails" phenomenon also arose from the Lost Notebook.This talk will be a personal account of highlights from this project and questions, yet to be answered, that arose from this decades long effort.

## François Bergeron: *Modules for rectangular Catalan combinatorics, and beyond ...*

Abstract:We propose explicit \(S_n\)-modules that explain the rich mixture of symmetric functions and combinatorics that has recently been studied in the context of rectangular (vs rational) Catalan combinatorics. Together with these modules, comes a much deeper understanding of how their different \(S_n\)-isotypic components relate one to the other. Furthermore, an interesting connection to the Macdonald Delta operators seems to be involved. This opens up a wide range of new problems in Enumerative and Algebraic Combinatorics.

## Mireille Bousquet Mélou: *Plane bipolar orientations and quadrant walks.*

(Joint work with Éric Fusy and Kilian Raschel)

Abstract:Our understanding of planar maps has evolved a lot since the early enumerative results of Tutte, obtained via a recursive approach in the sixties. Thirty years later, the simplicity of his formulae was at last understood at a combinatorial level, and the underlying bijections then paved the way to the study of large random maps, seen as metric spaces.For maps equipped with an additional structure, many questions remain open. In this talk, we consider planar maps equipped with a bipolar orientation, and show that they have a very rich combinatorial structure, related, among other topics, to lattice walks confined to cones. This allows us to count them, both recursively and bijectively, and to exhibit various universal properties of these structures.

## Mihai Ciucu: *Symmetries of shamrocks.*

Abstract:Hexagons with four-lobed regions called shamrocks removed from their center were introduced in their 2013 paper by Ciucu and Krattenthaler, where product formulas for the number of their lozenge tilings were provided. In analogy with the plane partitions which they generalize, we consider the problem of enumerating the lozenge tilings which are invariant under some symmetries of the underlying region. This leads to six symmetry classes besides the base case of requiring no symmetry. In this talk we present product formulas for two of these symmetry classes (namely, the ones generalizing cyclically symmetric, and cyclically symmetric and transpose complementary plane partitions).

## Theresia Eisenkölbl: *60th birthday of Christian Krattenthaler.*

## Tony Guttmann: *Combinatorial problems with stretched exponential asymptotics.*

Abstract:We look at a range of combinatorial problems where the growth of coefficients is of the form \(C.\mu^n\cdot \exp(-\alpha n^{\beta}) \cdot n^g,\) with \(\alpha > 0,\) \(0 < \beta < 1.\) Problems include some pattern-avoiding permutations, "pushed" random walks, "pushed" self-avoiding walks and interacting partially-directed walks. We will discuss, in a hand-waving way, how this stretched-exponential term arises, and give a numerical method for estimating the parameters \(\mu,\) \(\alpha,\) \(\beta\) and \(g.\) As an example, we give more precise asymptotics for the coefficients of \(Av(1324)\) pattern-avoiding permutations.

## Manuel Kauers: *Onsager's solution of the Ising model could have been guessed.*

Abstract:In the 1940s, Onsager found a closed form solution for the free energy in the 2D square Ising model without magnetic field. His formula is celebrated as one of the greatest scientific achievements of the 20th century. Although the derivation has been considerably simplified during the past decades, even the simplest derivation known today is not simple. In the talk, we will present a simple non-rigorous derivation of Onsager's formula starting from knowledge that was available to Onsager and using modern computer algebra, which Onsager of course did not have.This is joint work with Doron Zeilberger (arXiv:1805.09057).

## Thomas Müller: *A non-standard exponential principle for species.*

Abstract:We describe an exponential principle, which explains the exponential formulae occurring in Enumerative Combinatorics by relating them to a structural property of the corresponding species.

## Peter Paule: *Ramanujan's congruences modulo powers of 5, 7, and 11 revisited.*

Abstract:In 1919 Ramanujan conjectured three infinite families of congruences for the partition function modulo powers of 5, 7, and 11. In 1938 Watson proved the 5-case and (a corrected version of) the 7-case. In 1967 Atkin proved the remaining 11-family using a method significantly different from Watson's. In joint work with Silviu Radu (RISC) we set up a new algorithmic framework which brings all these cases under one umbrella. In this talk I will report on various new aspects of this setting. One aspect concerns a statement of Atkin who remarked that, in comparison with the 5 and 7-case, his proof for 11 is "indeed `langweilig', as Watson suggested." In our framework we find the 11-case particularly interesting.

## Tanguy Rivoal: *Hypergeometry and zeta values.*

Abstract:Hypergeometric identities and transformations play an important role in the study of the arithmetic nature of zeta values. I will present many classical Diophantine constructions, based on Padé type approximation of polylogarithms, to highlight this fact.

## Bruce Sagan: *Combinatorial interpretations of Lucas analogues.*

Abstract:The Lucas sequence is a sequence of polynomials in \(s,t\) defined recursively by \(\{0\}=0\), \(\{1\}=1\), and \(\{n\}=s\{n-1\}+t\{n-2\}\) for \(n\ge2\). On specialization of \(s\) and \(t\) one can recover the Fibonacci numbers, the nonnegative integers, and the \(q\)-integers \([n]_q\). Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of \(n\) in the expression with \(\{n\}\). It is then natural to ask if the resulting rational function is actually a polynomial in \(s\) and \(t\) and, if so, what it counts. Using lattice paths, we give combinatorial models for Lucas analogues of binomial coefficients as well as Catalan numbers and their relatives, such as those for finite Coxeter groups. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek.

## Xavier Viennot: *The essence of bijections: from growth diagrams to Laguerre heaps of segments for the PASEP.*

Abstract:The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems with a very rich underlying combinatorics in relation with orthogonal polynomials culminating in the combinatorics of the moments of the Askey-Wilson polynomials. I will begin with a tour of the PASEP combinatorial garden with many objects such as alternative, tree-like and Dyck tableaux, Laguerre and subdivided Laguerre histories, all of them enumerated by n!. Using several bijections relating these objects, Josuat-Vergs gave the most simple interpretation of the partition function of the 3 parameters PASEP in terms of permutations related to the moments of the Al-Salam-Chihara polynomials. This beautiful interpretation can be "explained" by introducing a new object called "Laguerre heaps of segments" having a central position among the several bijections of the PASEP garden. I will discuss some relations between these bijections and extract what can be called the "essence" of these bijections, some of them having the same "essence" as the Robinson-Schensted correspondence expressed with Fomin growth diagrams, dear to Christian.

## Ole Warnaar: *The Selberg integral and the AGT conjecture.*

Abstract:The Selberg integral is one of the most important hypergeometric integrals in all of mathematics. It has applications in many areas of mathematics and physics, including random matrix theory, number theory and conformal field theory. In a famous 2009 paper Alday, Gaiotto and Tachikawa conjectured a relation between conformal blocks in Liouville field theory and the Nekrasov partition function from \(\mathcal{N}=2\) supersymmetric gauge theory. One way to approach this conjecture is to compute Selberg integrals over products of Jack polynomials. In this talk I will report on some recent progress on this problem, as well as the many remaining issues, including the correct formulation of the problem in the language of algebraic combinatorics and special functions, free from the divergences and inconsistencies common in quantum field theory.

## Jiang Zeng: *Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions.*

Abstract:I will present Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. These results contain many previously obtained identities as special cases, providing a common refinement of all of them. Proofs of the main results will be outlined. This talk is based on joint work with Alan Sokal.

## Wadim Zudilin: *Creative microscoping.*

Abstract:It is well known that the limits of \(q\)-hypergeometric identities as \(q\to1\) recover the underlying 'ordinary' versions, and the latter normally serve as a motivation for discoverying the former. In my talk I will explain how the radial limits, as \(q\) tends to a root of unity (that is, at a '\(q\)-microscopic' level!), of the same \(q\)-hypergeometric identities can lead to the \(p\)-adic, aka congruence, counterparts of those ordinary versions. A typical example includes derivation, from a \(q\)-analogue of Ramanujan's formula

$$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1)=\frac{2\sqrt{3}}{\pi}, $$of the two supercongruences

$$ S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac{p-1}2\Bigr)\equiv p\biggl(\frac{-3}p\biggr)\pmod{ p^3}, $$valid for all primes \(p>3\), where \(S(N)\) denotes the truncation of the infinite sum at the \(N\)-th place and \((\frac{-3}{\cdot})\) stands for the quadratic character modulo 3.

The talk is based on joint work with Victor Guo.