Past events
Spring 2012
 May 2012
 Frédéric Paugam
 May 3 at 2pm, May 4 at 10am

 Functorial geometry and analysis on spaces of fields: definitions and examples
 Examples of action functionals: the fermionic particle, pure YangMills, YangMills with matter, general relativity
The working group is devoted to a series of seminars “Towards the mathematics of quantum field theory” by Frédéric Paugam, University Paris 7.
 Frédéric Paugam
 May 10 at 2pm, May 11 at 10am
 Frédéric Paugam
 May 24 at 2pm, May 25 at 10am

Algebraic tools for the structural study of nonlinear PDEs:
 Dmodules and Dalgebras. Differential Dgeometry
 The Ran space: local and chiral operations
 Frédéric Paugam
 May 31 at 2pm, June 1 at 10am

The classical Batalin–Vilkovisky formalism:
 The derived critical Dspace and gauge symmetries
 The classical master equation and the gauge fixing procedure
The working group is devoted to Dmodules, with five lectures by Pierre Schapira. The references are the book by Masaki Kashiwara on Dmodules and Microlocal Calculus, and the draft by Pierre Schapira From Dmodules to Deformation Quantization Modules (Ch 1,2).
 Pierre Shapira
 on March 15, 22, 29 and April 12, 19 at 2pm (Room B14)
 I will explain the basic notions of Dmodule theory in the analytic setting: construction and main properties of the sheaf of rings of differential operators on a complex manifold, characteristic variety of coherent Dmodules, operations on Dmodules (inverse and direct images), holomorphic solutions of Dmodules. If time allows it, we will have a glance to microdifferential operators and the link with deformation quantization. I will use freely the language of derived categories and sheaves, but there is no need to know anything on derived categories. On the other hand, some familiarity with basic sheaf theory and basic complex geometry is welcome.
Fall 2011
 Melchior Grützmann
 Htwisted Lie and Htwisted Courant algebroids and their cohomology
 We will introduce a new kind of algebroid similar to a Lie algebroid, but the Jacobi identity twisted by a 3form with values in the anchor map. Already Lie algebras permit the construction of nontrivial examples. We will introduce three kinds of cohomology theory, two in the language of dgsupermanifolds. Another class of examples occurs as Dirac structures in (splittable) twisted Courant algebroids a generalization of Hansen and Strobl's idea of twisting Courant algebroids with a 4form.
The working group started with a lecture by a guest of the team.
As from October, the working group has run a weekly research seminar on Covariant Field Theory. The lectures has been given by N. Poncin, they are mainly based on Frédéric Paugam’s recent works.
 1. Algebraic Geometry
 Generalized algebraic varieties, schemes, functor of points.
 2. Points and coordinates in Geometry and Physics
 ‘Point’ and ‘function’ viewpoints, Grothendieck topology, spaces: sheaves on a site, specific spaces: varieties and schemes, geometric constructions on spaces, spaces described by equations: ‘point’ approach and algebraic building blocks.
 3. Schemes relative to a symmetric monoidal category
 Complements on closed monoidal categories, relative schemes, relative differential calculus.
 4. Applications to Classical Mechanics and Field Theory
 International workshop
 Giuseppe Bonavolontà and Norbert Poncin organized an international workshop on Covariant Field Theory from 6 to 8 December 2011.