The Working Group "Algebraic Topology, Geometry and Physics" is a weekly meeting of the research team of Prof. Norbert Poncin. The aim is to present both, research works and surveys of mathematical areas of common interest.

Meetings

Every Thursday/Friday in the room B.23 campus Kirchberg.

Organizer

Please feel free to contact Jean-Philippe Michel for any questions and if you want to be added to the mailing list.

Present sessions

May 2012

The working group is devoted to a series of seminars ‘Towards the mathematics of quantum field theory’ by Frederic Paugam, University Paris 7.

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 Yang-Mills, Yang-Mills with matter, general relativity
    • 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 non-linear PDEs:
  • D-modules and D-algebras. Differential D-geometry
  • 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 D-space and gauge symmetries
  • The classical master equation and the gauge fixing procedure
    • March 2012 - April 2012

    The working group is devoted to D-modules, with five lectures by Pierre Schapira. The references are the book by Masaki Kashiwara on D-modules and Microlocal Calculus, and the draft by Pierre Schapira From D-modules 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 D-module 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 D-modules, operations on D-modules (inverse and direct images), holomorphic solutions of D-modules. 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.

    Past sessions

    Fall 2011

    The working group started with a lecture by a guest of the team.

    Melchior Grützmann
    H-twisted Lie and H-twisted 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 3-form with values in the anchor map. Already Lie algebras permit the construction of non-trivial examples. We will introduce three kinds of cohomology theory, two in the language of dg-supermanifolds. 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 4-form.

    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

    The semester activities culminated with a workshop on Covariant Field Theory, with the following speakers:

    G. Barnich
    BV-cohomology (II)
    J. Grabowski
    Tulczyjew triples in mechanics and field theory
    A. Kotov
    Jet spaces and superfields (II)
    G. Moreno
    Mini-course: Horizontal cohomology (III)
    L. Vitagliano
    Minicourse: Covariant phase space (III)
    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) has given a lecture course in Fall 2010 on Gauge Field Theory: locality, symmetries and BV formalism. One of the references is the book of 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 algebroïd and BV formalism
    Gauge systems as Lie algebroids, homological perturbation theory, BV differential, antibracket, master action, examples.