CM - Operads in combinatorics - M2 informatique fondamentale

Abstract

Informally, an operad is a space of operations having one output and several inputs that can be composed. Each operad leads to the definition of category of algebras. This theory offers a tool to study situations wherein several operations interact with each others.

This lecture begins by presenting some elementary objects of algebraic combinatorics: combinatorial classes and combinatorial algebras. We introduce then (non-symmetric) operads and study some tools allowing to establish presentations by generators and relations of operads. Koszul duality in non-symmetric operads is an important part of this theory which shall be presented.

We end this lecture by reviewing some generalizations: colored operads, symmetric operads, and pros. We shall also explain how the theory of operads offers a tool to obtain enumerative results.

Lectures

Here are the main presented notions:

• Lecture 1, January 16, 2019: combinatorial collections; operations on combinatorial collections.


• Lecture 2, January 23, 2019: treelike structures; syntax trees; operations on syntax trees.


• Lecture 3, January 30, 2019 : rewrite systems on trees.


• Lecture 4, February 6, 2019 : exercices on rewrite systems on trees (Tamari order, Motzkin trees, etc.).


• Lecture 5, February 13, 2019 : series and polynomials on combinatorial collections; biproducts; associative algebras, coassociative algebras; dendriform algebras.


• Lecture 6, February 20, 2019 : associative algebras of permutations (shuffle, convolution, super-shuffle); pre-Lie algebras and pre-Lie algebras of rooted trees.


• Lecture 7, February 27, 2019 : operads; algebras over operads; presentations by generators and relations; proving presentations.


• Lecture 8, March 6, 2019 : evaluation: presentation of a research paper.