Professor Norbert Poncin's research group “Algebraic Topology, Geometry and Physics” holds weekly meetings whose aim is to present both original research work and surveys of mathematical areas of common interest.

### Time and venue

Unless otherwise specified, the usual meeting time is 2:15 to 3:45 PM on Thursdays

in the library room of building G, Campus Kirchberg.

### Calendar

#### Fall 2015

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
##### Past events
Wednesday, December 9 - Friday, December 11, 2015
Higher Geometry and Field Theory Conference at the University of Luxembourg
Organized by Stephen Kwok, Norbert Poncin, and Vladimir Salnikov.
December 3, 2015, from 2:15 to 3:45 PM - Salle des conseils, CK
Alessandro Valentino (Max Planck Institute for Mathematics, Bonn)
Central extensions of mapping class groups from characteristic classes
I will discuss a functorial construction of extensions of mapping class groups of smooth manifolds which are induced by extensions of (higher) diffeomorphism groups via the group stack of automorphisms of manifolds equipped with higher degree topological structures. The problem of constructing such extensions arises naturally in the study of topological quantum field theories, in particular in 3d Chern-Simons theory. Based on joint work with Domenico Fiorenza and Urs Schreiber.
Tuesday, November 24, 2015, from 4:00 to 5:00 PM - Room B02, CK ~ Joint session with the General Mathematics Seminar
Noncommutative analogues of moduli spaces via brick manifolds
I shall talk about a remarkable series of algebraic varieties that resemble the Deligne-Mumford compactifications of moduli spaces of curves of genus zero with marked points. They admit three equivalent descriptions: as "brick manifolds" recently defined by Escobar, as toric varieties of Loday's realisations of associahedra, and as De Concini-Procesi wonderful models of certain subspace arrangements. The talk will introduce, on a level accessible to a general mathematics audience, these three set-ups from scratch, and outline remarkable properties behind those spaces. It is based on a joint work with Sergey Shadrin and Bruno Vallette (http://arxiv.org/abs/1510.03261).
Tuesday, November 17, 2015
Research Seminar
Tuesday, November 10, 2015, from 4:00 to 5:00 PM - Room B02, CK ~ Joint session with the General Mathematics Seminar
Rita Fioresi (University of Bologna, Italy)
Representations of compact supergroups and SUSY structures
In this talk we present some results on the Peter-Weyl theorem for the special examples of $S^{1|1}$ and $S^{1|2}$ and the connection with SUSY-structures.
November 5, 2015, from 2.15 to 3.15 PM - Room B27, CK
Floriar Schätz (University of Luxembourg)
Variations on the BRST construction
The classical BRST construction provides a homological framework for symplectic reduction. After recalling how this works in the simplest setting, I will outline a generalization - known as the BFV construction - to arbitrary coisotropic submanifolds and its use in deformation theory. I will finish with some thoughts on Poisson submanifolds.
October 27, 2015, from 2.15 to 3.15 PM - Room B27, CK
Graded geometry in gauge theories and beyond (part 2: multiple gradings and SUSY)
In this second part of the talk I am going to address the question of generalizing the results of the first part to "supersymmetric" theories. This is mostly work in progress related to introduction of multiple gradings in the context of the Poisson sigma model and the Chern-Simons theory. The global goal is to approach a clean construction of an appropriate multigraded analog of the AKSZ procedure. I will also provide some (rather toy model) examples.
October 20, 2015, from 2:00 to 3:00 PM ~ Joint session with the Algebra, Geometry and Quantization Seminar
Graded geometry in gauge theories and beyond (part 1)
We study graded geometric constructions appearing naturally in the context of gauge theories. We introduce the language of Q-bundles convenient for description of symmetries of sigma models. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q-manifolds introducing thus the concept of equivariant Q-cohomology.
This notion turns out to be useful for analysis of such theories as the (twisted) Poisson sigma model and the Dirac sigma model. We obtain these models by a gauging-type procedure of the action of a group related to Lie algebroids and n-plectic manifolds. We also show that the Dirac sigma model is universal in space-time dimension 2. On top of applications to gauge theories (time permitting) I will comment on a possible definition of equivariant cohomology for Courant algebroids.
This is a joint work with Thomas Strobl, and in part with Alexei Kotov.

#### Spring 2015

May 12, 2015
Yannick Voglaire (University of Luxembourg)
Differentiable stacks from the point of view of Lie groupoids - Part 2
In this second talk, I will continue to describe the relation between differentiable stacks and Lie groupoids, focusing this time on Lie groupoids. I will provide explicit examples and, if time permits, I will explain the relation to C*-algebras.
April 21, 2015
Alessandro Zampini
How to define a Dirac operator following Kähler's approach
April 7, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 5
We will conclude our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will continue our discussion of super Fermat theories and show that every ordinary Fermat theory may be extended canonically to a super Fermat theory.
March 31, 2015
François Petit (University of Luxembourg)
Stable and presentable infinity-categories
The theory of presentable and especially presentable and stable $\infty$-categories provides a very powerful framework. For instance, stable presentable categories form a monoidal category, there is also an adjoint functor theorem and a version of Brown representability theorem for such categories. In this talk, I will present, following Lurie, a brief overview of the theory of stable and presentable categories from the viewpoint of the user. If time permits, I will also discuss spectra and stabilization.
March 24, 2015
Alessandro Zampini
Spin geometry and Dirac operators, part 2
Monday, March 23, 2015, from 4:00 to 5:30 PM
Alessandro Zampini
Spin geometry and Dirac operators, part 1
March 17, 2015
Yannick Voglaire (University of Luxembourg)
Differentiable stacks from the point of view of Lie groupoids
In this introductory talk, I will introduce differentiable stacks and show how they are presented by Lie groupoids. I will describe generalized morphisms between Lie groupoids and the resulting notion of Morita equivalence. I will then explain in which precise (bicategorical) way isomorphic differentiable stack correspond to Morita equivalent Lie groupoids.
March 10, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 4
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will discuss nilpotent extensions in the context of Fermat theories, superalgebras, and super Fermat theories.
March 3, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 3
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will continue to discuss the development of differential calculus in the context the Fermat theories of Dubuc-Kock.
February 17, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 2
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will present further background material on the Fermat theories of Dubuc-Kock, and the development of differential calculus in the context of such theories.
Thursday, January 29, 2015, from 2:15 to 3:45 PM
Stephen Kwok
On theories of superalgebras of differentiable functions, 1
Dubuc and Kock developed the concept of Fermat theories, theories of commutative algebras for which infinitely differentiable functions may be evaluated on elements. These theories arose in the development of models for synthetic differential geometry, but have more recently played a key role in the development of models for derived differential geometry.
In this series of talks, we will discuss the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this first talk, we will present background material on Lawvere algebraic theories, and the Fermat theories of Dubuc-Kock.
January 16-17, 2015
Informal meetings on the occasion of the visit of Luca Vitagliano (University of Salerno)

#### Fall 2014

December 2, 2014
Damjan Pištalo
The BRST construction, Part II
This is the second part of last week's talk. We will combine the Koszul-Tate and Longitudinal differentials to construct the BRST complex.
November 25, 2014
Damjan Pištalo
The BRST construction, Part I
In my last talk, I defined the longitudinal differential. I will now show that its zero homology is equal to the space of observables. I will then define the Koszul-Tate resolution as a semifree resolution of $C^\infty(\Sigma)$, where $\Sigma$ is the constraint surface. Combining Koszul-Tate and Longitudinal differential, I will construct the BRST complex.
November 18, 2014
Damjan Pištalo
Constrained Hamiltonian systems, III
This is the third talk in our series. After reviewing the symplectic structure on the phase space induced by the Poisson bracket, we will show that zero vector fields of the induced two form on the constraint surface are generators of infinitesimal gauge transformations. Moreover, such vector fields satisfy Frobenius' theorem on the constraint surface. This allows us to define the de Rham complex mentioned previously.
November 11, 2014
Damjan Pištalo
Constrained Hamiltonian systems, II
This is the second part of last week's talk.
November 4, 2014
Damjan Pištalo
Constrained Hamiltonian systems, I
In gauge theory, equations of motion contain arbitrary functions of time. This leads to relations (constraints) between the canonical variables $p$ and $q$ in the Hamiltonian formalism, restricting the motion to a submanifold of the phase space (the constraint surface). Constraints are divided into first and second class. The existence of first class constraints corresponds to the existence of gauge symmetries, which is the case when more than one set of canonical variables describe the same physical state. Vector fields generating infinitesimal gauge transformations satisfy Frobenius' theorem on the constraint surface. This enables us to define something similar to de Rham complex over their “duals”. At the end, I will establish a 1-1 correspondence between these vector fields and first class constraints, that will be crucial for the antighost-ghost correspondence needed in the BRST formalism.

#### Spring 2014

27 May 2014
Alessandro Zampini
Quantumness and classicality, 2
This is the second part of last week's talk.
20 May 2014
Alessandro Zampini
Quantumness and classicality, 1
In this talk I shall describe how the geometry of C*-algebras and Jordan algebras can be useful to investigate whether a physical system has a quantum vs. a classical description.
13 May 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 6
Starting with a brief review of what was done in the last talk and making more precise some of the notions we introduced last week, today we will finally deal with the definition of a stack in the sense of Toen and Vezzosi. We will then give an idea of the reason why they can be used as the building blocks for derived algebraic geometry.
29 Apr 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 5
Simplicial Presheaves and Model Sites: The purpose of the fifth talk of our series will be to deal with simplicial presheaves on a model category and to define model topologies. We will then introduce pre-stacks and stacks on model sites.
8 Apr 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 4
Towards homotopy theory for model categories: In our fourth meeting, we will deal with the small object argument and finish talking about the example of the category of nonnegatively graded chain complexes. We will then end with a brief sketch of the definition of the homotopy category of a model category.
1 Apr 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 3
In our third talk, we will introduce model categories, following Quillen. After presenting the basics of model category theory, we will focus mostly on the example provided by the category of non-negatively graded chain complexes of modules over an associative ring with unit.
18 Mar 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 2
Motivated by the representability criterion we stated last time, we will deal with Grothendieck topologies and sheaves (see last part of last week's abstract). Time permitting, we will then start talking about model categories.
11 Mar 2014
Gennaro Di Brino
Foundations of Homotopical Algebraic Geometry, 1
In this series of talks we will use the various formulations of the category-theoretical Yoneda lemma to explore Homotopical Algebraic Geometry (HAG 1 and 2), the foundational work of Toen and Vezzosi.
The classical setting: We plan to start from a standard representability proof for a (contravariant) functor from the category of schemes to that of sets. We will see how the original lemma of Yoneda comes into play in that setting and we will use the classical representability criterion as the starting point for the introduction of Grothendieck topologies on a category, and sheaves on Grothendieck sites.
25 Feb 2014
Benoît Jubin
Super Harish-Chandra pairs and graded Lie algebras
I will briefly recall the notions of super Lie groups and their morphisms, the construction of the super Lie algebra of a super Lie group, and super Lie group actions. Then, I will introduce super Harish-Chandra pairs and their morphisms. I will construct a functor from the category of super Lie groups to that of super Harish-Chandra pairs, and a quasi-inverse of that functor, proving the equivalence of these two categories with a constructive proof. These constructions are due to Kostant, Koszul, Vishnyakova, Carmeli–Fioresi, among others. Time permitting, I will say how this can be adapted to the $\mathbb{Z}$-graded case.

#### Fall 2013

17 Dec 2013
Alessandro Zampini
The Hochschild–Kostant–Rosenberg theorem
The aim of the talk is to describe how the differential calculus on a topological space can be given a homological interpretation, and then to show how this allows (Connes' approach) to extend such a notion to non-commutative spaces.
10 Dec 2013
Benoît Jubin and Norbert Poncin organized an international workshop on Higher Lie Theory from 9 to 11 December 2013.
3 Dec 2013
Kyosuke Uchino
26 Nov 2013
Jian Qiu
Integration of Lie infinity algebras (Henriques' method)
In contrast to the integration method due to Getzler, who designed an iteration procedure to solve the Maurer–Cartan (MC) equation directly, Henriques placed a filtration on the Lie infinity algebra, that allows him to solve the MC equation in successive approximations. His method dispenses from the nilpotency restriction of Getzler, yet on the other hand the filtration is not valid when the Lie algebra has terms in the positive grading.
19 Nov 2013
Benoît Jubin
DG Lie groups and DG Lie algebras, II
12 Nov 2013
Benoît Jubin
DG Lie groups and DG Lie algebras, I
After a review of graded manifolds, I will introduce graded Lie groups and show how to construct their tangent graded Lie algebras, using left-invariant vector fields. Then, I will recall the notions of homological vector field and dg manifold, and I will introduce dg Lie groups and show how to associate to each of them a dg Lie algebra. In the process, I will survey the theory of graded Hopf algebras, and, as a warm-up, I will show how multiplicative vector fields on Lie groups give rise to differentials on their Lie algebras (via the Van Est isomorphism). I will hint at the inverse constructions (the “integration problem”) of all these steps.

#### Spring 2013

11 Jul 2013
Mathieu Stiénon
Atiyah classes and homotopy algebras
The Atiyah class of a holomorphic vector bundle $E$ is the obstruction to the existence of a holomorphic connection on $E$. A theorem of Kapranov states that, for a complex manifold $X$, the Atiyah class of $T_X$ makes $T_X[-1]$ into a Lie algebra object in the derived category $D(X)$. Furthermore, Kapranov proved that, for Kaehler manifolds, this Lie algebra structure stems from an $L_\infty$ algebra structure on $\Omega^{0,*}[-1](T_X)$. In this talk, we will show how Kapranov's theorems can be extended to the more general setting of Lie pairs of algebroids so as to produce new homotopy algebras.
28 May 2013
Alessandro Zampini
Differential calculi on quantum groups
15 May 2013
Shuffle operads and their not yet discovered generalisations (II)
14 May 2013
Shuffle operads and their not yet discovered generalisations (I)
Shuffle operads can be used to approach questions about symmetric operads in a fairly efficient algorithmic way. The key idea behind the notion of a shuffle operad is to no longer allow the symmetric groups to act directly on operations, but only keep some of the indirect actions on results of compositions. I shall remind how that works, and then discuss two generalisations of symmetric operads for which it would be desirable to find "shuffle shadows", namely cyclic operads and wheeled operads. In the first case, I shall outline a solution, in second one, explain potential approaches to the problem.
07 May 2013
Alessandro Zampini
What is a quantum group? (II)
30 Apr 2013
Alessandro Zampini
What is a quantum group? (I)
I shall introduce the notion of quantum group and describe particularly the quantum deformation of the classical Lie group $SU(2)$.
From 12 Mar to 16 Apr Professor Pierre Schapira gave the minicourse “Introduction to D-modules on Ran spaces”. It consisted of 5 lectures. The schedule and the list of references can be downloaded here.
16 Apr 2013
Pierre Schapira
Ran spaces, 2-limits of categories, $\mathcal{D}$-modules on Ran spaces, a glance to the chiral product and to non-linear equations (II)
9 Apr 2013
Pierre Schapira
Ran spaces, 2-limits of categories, $\mathcal{D}$-modules on Ran spaces, a glance to the chiral product and to non-linear equations (I)
26 Mar 2013
Pierre Schapira
Internal operations $\mathcal{Hom}$ and $\otimes$ and external operations (inverse and direct images)
19 Mar 2013
Pierre Schapira
Characteristic variety, holonomic modules
12 Mar 2013
Pierre Schapira
Construction and main properties of the sheaf of rings $\mathcal{D}$ of differential operators, De Rham and Spencer complexes
01 Mar 2013
Jian Qiu
Lie theory for nilpotent $L_\infty$-algebras (IV)
26 Feb 2013
Jian Qiu
Lie theory for nilpotent $L_\infty$-algebras (III)
The simplicial set of MC elements serves as integration object. The main goal of the talk is to show that it is a Kan simplicial set, i.e. if the MC elements are defined on a horn of a simplex, one can always find an extension to the entire simplex. Furthermore, one can impose a gauge condition and show that to a certain extent the extension is unique, so that one actually gets a (weak) l-group(oid). The extension formula also provides a generalization of the CBH formula.
22 Feb 2013
Benoît Jubin
Lie theory for nilpotent $L_\infty$-algebras (II)
19 Feb 2013
Benoît Jubin
Lie theory for nilpotent $L_\infty$-algebras (I)
This is a presentation in two parts of Getzler's Lie theory for nilpotent $L_\infty$-algebras. I will give some motivation with the example of DGLAs and say in what sense we want to integrate them. Then I will review the notions of nerve of a category, of Kan complex and of higher groupoid. I will present Dupont's explicit proof of the simplicial De Rham theorem used in the sequel. I will finally introduce nilpotent $L_\infty$-algebras and their Maurer–Cartan simplicial sets, and show that they are Kan complexes. A specific subset of the Maurer–Cartan simplicial set can be considered as the integration of the given $L_\infty$-algebra, as will be explained next week.
12 Feb 2013
Jian Qiu
Derived symplectic geometry
The talk is a report on a mini course given during the Winter School 2013 in Les Diablerets. I start with some examples of intersection problems to motivate the idea of 'derivation' — see http://streamer.cit.utexas.edu/math-grasp/lurie.html. Then I sketch the notions of forms, vector fields, and Lagrangian structures in derived geometry, and provide examples.

#### Fall 2012

30 Nov 2012
Simon Covez
The local integration of Leibniz algebras
The goal of this talk is to present on an example how a Leibniz algebra integrates into a local Lie rack.
Nov 2012
Jumpei Nogami
Series of six lectures on Higher Categories, Higher Algebra, and Derived Algebraic Geometry.
30 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), III
Integrability: smooth theory. Application to the integrability of Poisson manifolds.
26 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), II
A few more examples: Atiyah sequence, Poisson manifolds. Connections on Lie algebroids. Representations and the general Lie algebroid. Integration of Lie algebroids: topological theory. The s-simply-connected Lie groupoid associated to a Lie groupoid. Uniqueness for the integrability problem. The “path space” (Duistermaat–Kolk) proof that any Lie algebra is the Lie algebra of a simply-connected Lie group. Notions of G-path, A-path, and A-homotopy. Construction of the Weinstein groupoid of a Lie algebroid. Monodromy groups of a Lie algebroid. Exponential map.
16 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), I
Lie groupoids: definition and basic properties (s-fibers, isotropy groups, orbits, subgroupoids, morphisms). Examples (Lie group, pair groupoid, gauge groupoid, general linear groupoid, action groupoid, fundamental groupoid of a manifold and of a foliation). Representation of a Lie groupoid. Lie algebroids: definition and basic properties (isotropy Lie algebras, orbits, morphisms). Construction of the Lie algebroid of a Lie groupoid. Examples of this construction: Lie algebra, tangent bundle, action Lie algebroid.
09 Oct 2012
Jian Qiu
Model Categories II
This is a continuation of my previous seminar. I will explain how to formulate homotopy theory in a model category. This involves path and cylinder objects, as well as (co)fibrant replacements, and leads us naturally to derived functors. As an example, I work out, using the language of derived categories, the derived zero locus of a section in the purely algebraic setting.
02 Oct 2012
Tiffany Covolo
Higher trace and Berezinian of matrices over Clifford algebras
I will report on a recently published joint work with N. Poncin and V. Ovsienko. I will first introduce some notions of graded algebra (a generalization of superalgebra), then I will focus on the $(\mathbb{Z}_2)^n$ - graded version of the trace and of the Berezinian.
25 Sep 2012
Jian Qiu
Model categories
This talk is meant to be an introduction to model category theory. We start from the axioms of model categories, and illustrate them with two of the most prominent examples, topological spaces and chain complexes. Then we take a look at the formulation of homotopy in the model category and talk briefly about Quillen functors and other model categories.
18 Sep 2012
Benoît Jubin
Courant algebroids
I will show how Dirac structures and the Courant bracket on $TM\oplus T^*M$ appear as a common generalization of presymplectic and Poisson structures (after Courant, Weinstein). Then I will define general Courant algebroids and show how they arise as the double objects of Lie bialgebroids, in analogy to the Drinfeld double of a Lie bialgebra (after Liu, Weinstein, Xu). I will also discuss Severa's classification of exact Courant algebroids and the supergeometric interpretation of Courant algebroids (after Severa and Roytenberg).

#### Spring 2012

May 2012

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 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.

#### 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
International workshop
Giuseppe Bonavolontà and Norbert Poncin organized an international workshop on Covariant Field Theory from 6 to 8 December 2011.

#### 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.
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.
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.