### Past events

#### Spring 2011

The working group has been devoted in Spring 2011 to an introduction to operads, based on the draft "Algebraic operads" by Jean-Louis Loday and Bruno Valette. The talks have been given by Norbert Poncin and Ashis Mandal.

- 1. Preliminaries
- Representations of finite groups, coalgebras, homological algebra.
- 2. Algebraic operads
- Categories, higher categories, multicategories, operads, classical and functorial definitions of operads, examples, algebras over operads, free operad and cooperad.
- 3. Bar-cobar resolution
- Bar and cobar constructions, twisting and Koszul morphisms, BC-resolution.
- 4. Koszul duality for associative algebras
- Quadratic algebras and coalgebras, Koszul dual algebra and coalgebra, Koszul algebras, Koszul resolution.
- 5. Operadic bar-cobar resolution
- Infinitesimal composite, differential graded operads and cooperads, operadic bar and cobar constructions, operadic twisting and Koszul morphisms, BC-resolution.
- 6. Koszul duality for operads
- Quadratic operads and cooperads, Koszul dual operad and cooperad, Koszul operads, operadic Koszul resolution.
- 7. Homotopy algebras over quadratic Koszul operads
- Transfer theorem, A-infinity algebras, Stasheff polytope, operads A-infinity and As-infinity.

#### Fall 2010

Glenn Barnich (Free University of Brussels) gave a course on Gauge Field Theory: locality, symmetries and BV formalism. One of the references is the notes The variational bicomplex by Ian M. Anderson. The sessions have been the following:

- 1.Jet-spaces and horizontal complex
- Derivatives as coordinates, total and Euler-Lagrange derivatives, local functions, local functionals.
- 2. Jet-spaces and horizontal complex
- Horizontal complex, local exactness.
- 3. Jet-spaces and horizontal complex
- Remarks on the variational bi-complex and the inverse problem.
- 4. Dynamics
- Equations of motion as a surface, Noether identities.
- 5. Dynamics
- Homological Koszul-Tate resolution, characteristic cohomology.
- 6. Symmetries
- Generalized vector fields, prolongation, evolutionary vector fields, symmetries of the equations, variational symmetries.
- 7. Symmetries
- Gauge symmetries, generalized Noether theorems.
- 8. Gauge algebroid and BV formalism
- Gauge systems as Lie algebroids, homological perturbation theory, BV differential, antibracket, master action, examples.