INTRODUCTION TO ALGEBRAIC OPERADS
WiSe 2022/2023 Humboldt University

Lecture notes                              

Exercice sheets:  1  2  3  4  5  6               

HOMEWORK 1 (DUE 11:59PM DECEMBER 21)

Final exam:  one oral exam of 30 min, either on the 22/02 or the 06/04

Initially introduced in the sixties to tackle several problems in algebraic topology, the theory of algebraic operads has experienced spectacular developments since the nineties, and is now a central tool in various fields of modern mathematics: deformation quantization, string topology, Morse theory, symplectic topology or algebraic geometry.

It aims at defining an appropriate framework to study algebraic structures and their homotopy theory. The paradigm is to see the collection of operations encoding a specific algebraic structure, as well as the relations they have to satisfy, as an algebraic entity on its own: this algebraic entity is called an operad. Put differently, an operad P encodes a category of P-algebras. For instance, the operad denoted As will define the classical category of associative algebras. One can then study the properties of the category of P-algebras by studying directly the properties of the operad P, using systematic methods.

The current plan for these lectures goes as follows:
- Provide a general introduction to the notion of operad and the various constructions associated to it
- Study several classical examples of differential graded operads and of topological operads
- Present Koszul duality theory of algebras and dg operads
- Apply this theory to the study of homotopy theory of different categories of algebras (associative algebras, Lie algebras, BV/Gerstenhaber algebras)
- If time permits, sketch several current problems of research on operads