Algebraic Topology, Geometry and Physics

Seminar at Mathematics Research Unit

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.

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