Abstract: Maps describe the embedding of graphs on orientable surfaces, and are a classical subject in enumerative and algebraic combinatorics. They can be encoded as walks in symmetric groups, and as such they can be enumerated via representation theory of these groups. This leads to an explicit expression of certain generating series of maps in terms of Schur functions -- or equivalently, as generating functions of certain tableaux. This link (maps/tableaux)  has extremely deep consequences (such as the KP hierarchy for map generating functions) but has remained mysterious at the combinatorial level. 

The final goal of this course is to present a recent proof of this map-tableau correspondence that is independent of representation theory. It consists in showing that both sides of the correspondence are solutions of the "Tutte equations" and generalizations of it. This relies, on one side, on decompositions of maps via  "root deletion" procedures, and on the other side, on classical tableaux rules such as Murnaghan-Nakayama. 

These lectures aim to be pedagogical and they will be a pretext to discover beautiful families of combinatorial objects such as maps, tableaux, or symmetric functions! 

The lectures are based on my paper with Maciej Dołęga (cf arxiv:2004.07824) and they can serve as an introduction to it. However, while most of the paper deals with non-oriented maps and Jack polynomials, the talk will remain at the level of oriented maps and Schur functions.

Exercises: www.irif.fr/~chapuy/exercisesBedlewo.pdf

Solutions to exercises: https://www.irif.fr/~chapuy/exercisesBedlewo-solutions.pdf

Abstract: Determinants play an important role in several domains of enumeration, in particular in the enumeration of tilings and of paths, the two being intimately connected. The goal of this series of lectures will be, first, to provide a survey of results and techniques that are available for the evaluation of determinants, second, to explain the significance of determinant evaluations in the enumeration of tilings and paths, and, third, to illustrate these theoretical findings by going through some not so recent and some more recent examples in the enumeration of (rhombus and domino) tilings.

Material for the course: https://www.mat.univie.ac.at/~kratt/bedlewo/

Abstract: Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives on their own. The flagship hook-length formula counts the number of Standard Young Tableaux, which also give the dimension of the irreducible Specht modules of the Symmetric group. The beautiful Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, they became applicable in Probability and Statistical Mechanics and Computational Complexity Theory. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and ``combinatorial interpretations''. 

In the opposite direction,  the 85 year old open problem on Kronecker coefficients of the Symmetric group lead to the disprove of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail. This, as well as the connections to Integrable Probability/Statistical Mechanics, prompt the understanding of the asymptotics of such representation theoretic quantities (structure constants) in various regimes. 

In this summer course we will start with basic Algebraic Combinatorics, and turn to the more recent developments towards Computational Complexity Theory, as well as asymptotics.