# 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;
dendriforme 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.