Algebraic Operads, Koszul duality and Gröbner bases: an introduction

The lecture series aim to offer a gentle introduction to the theory of algebraic operads, starting with the elements of the theory, and progressing slowly towards Koszul duality theory, Gröbner bases, and higher structures. We hope for the course to pave a roadway for participants to later explore more advanced themes independently: participants will be encouraged to read (parts of) accessible research articles and present them during the last sessions of the lecture series.

Operads appeared during the 60/70s in the work of algebraic topologists and algebraists and then reappeared during the early 90s, in their `renaissance' period. They have proven to be indispensable tools to state and prove results in the areas of algebraic topology, category and model theory, homotopy theory, real and complex geometry, mathematical physics, algebra and deformation theory, homotopy invariant structures, and graph complexes.

News

The course is now finished.

Lecture Notes

A draft version will be kept and updated in this link.

Current version: February 27th, 2022.

Lectures

The following is the final schedule for the 12 lectures of the course.

• Lecture 0. Motivation, (pre)history and background material.
• Lecture 1. Σ-sequences, composition product. Unbiased approach.
• Lecture 2. Category of operads. Rooted trees and free operads.
• Lecture 3. Presentations. Quadratic operads. Three graces and their kin.
• Lecture 4. DG sequences. Koszul dual of a quadratic operad.
• Lecture 5. Shuffle sequences and operads. Applying the forgetful functor.
• Lecture 6. Path sequences, leaf permutations and monomial orders.
• Lecture 7. Normal forms, long division and Gröbner bases.
• Lecture 8. S-polynomials, Diamond Lemma and Buchberger's Algorithm.
• Lecture 9. The bar and cobar constructions.
• Lecture 10. The 0th syzygy homology. Koszul complexes.
• Lecture 11. The numerical criterion. Monomial operads.
• Lecture 12. Filtrations and rewriting methods. Distributive laws.

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Exercises

These are incorporated in the lecture notes.

Literature

• Algebraic Operads. Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Springer-Verlag Berlin Heidelberg.
• Algebraic Operads: an algorithmic companion. Murray R. Bremner and Vladimir Dotsenko, Chapman and Hall/CRC.
• Koszul duality for operads. Victor Ginzburg and Mikhail Kapranov.
• Operads and PROPs. Martin Markl.
• Operads in Algebra, Topology and Physics. Martin Markl, Steve Shnider, and Jim Stasheff.
• Algebra+Homotopy=Operad. Bruno Vallette.
• Distributive Laws and Koszulness. Martin Markl.
• Quillen homology for operads via Gröbner bases. Vladimir Dotsenko and Anton Khoroshkin.
• Gröbner bases for operads. Vladimir Dotsenko and Anton Khoroshkin.
• Free resolutions via Gröbner bases. Vladimir Dotsenko and Anton Khoroshkin.

More resources

In this link you can find two blogposts introducing operads by Tai-Danae Bradley, which are very nicely illustrated and explained.