Let $(M,\omega)$ be a connected symplectic manifold, $\delta:\Omega^{\bullet}(M)\to\Omega^{\bullet-1}(M)$ the canonical homology operator, and $H_{\bullet}^{can}(M)$ the canonical homology groups.
In the first part of the mini-course we will explain why $\Omega^1(M)/\delta\Omega^2(M)$, endowed with the Lie bracket $[[\alpha],[\beta]]=[\delta\alpha\cdot d\delta\beta]$, is the universal central Lie algebra extension (by $H_1^{can}(M)$) of the Poisson Lie algebra $C^\infty(M)$ if $M$ non-compact, resp. of the Lie algebra of Hamiltonian vector fields if $M$ compact.
In the second part we will show how the Lie algebra $\Omega^1(M)/\delta\Omega^2(M)$ extends to an $L_\infty$-algebra structure on the canonical homology complex. The extension procedure is inspired by a similar approach for multisyplectic manifolds developed by Christopher Rogers, but is combinatorially more intricate and relies on results from symplectic Hodge theory.

We will explain the notion of abstract non-commutative calculus
and its relations to formality.
We will also describe some of the constructions involved in
concrete computations and some applications in analysis and deformation theory.

For certain maps π between foliated manifolds it is induced a morphism of their holonomy groupoids. The objective of this talk is to prove properties of this assignment when π satisfies a surjectivity property, and thus can be regarded as quotient map. We will discuss certain examples where π can be seen as a quotient map of a Lie 2-group actions on the holonomy groupoid.

Quotient spaces behaves different depending on which groupoid you decide to model it. Groupoids that are Morita equivalent give the same smooth behavior in their respective quotient space therefore the classes of Morita equivalent groupoids give the different models for the quotient space.
A natural example of quotient spaces are the ones given by a singular foliation on a manifold. In this case we have a canonical groupoid to model it, the holonomy groupoid.
This talk gives an introduction on singular foliations, it explains the construction of the holonomy groupoid of a singular foliation and discusses the definition of Morita equivalence.

The theory of G-structures provides a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart'' of symplectic geometry - does not fit naturally into this picture. In this talk, after a quick review on the very basics of G-structures (with examples), I will introduce the notion of a homogeneous G-structure, which encompasses contact structures, as well as some other interesting examples appeared in the literature.

In this short course we will focus on differential graded Lie algebras and their role in deformation theory, with particular attention to deformations of manifolds and pairs (manifold, sub manifold) and (manifold, vector bundle). More precisely, we would like to cover the following arguments:

Introduction to functor of Artin rings and moduli spaces.

Functors associated with DGLAs and examples.

Thom-Whitney totalisation and applications.

Homotopy abelian DGLAs and examples.

Deligne Groupoid and L_{∞} algebras.

Introduction to functor of Artin rings and moduli spaces.

Functors associated with DGLAs and examples.

Thom-Whitney totalisation and applications.

Homotopy abelian DGLAs and examples.

Deligne Groupoid and L

We will discuss deformations of symplectic groupoids, and how the Bott-Shulman-Stasheff (BSS) double complex shows up controlling infinitesimal deformations.
We will then see a map relating the differentiable cohomology to the deformation cohomology of a symplectic groupoid.
This talk is based on joint work with Ivan Struchiner and Cristian Cárdenas.

The present talk is a survey of part of recent joint work with Mauro Spera (arXiv: 1805.01696), in which we investigated some connections between multisymplectic geometry and knot theory.

A connection between these two topics can be established via mechanics of ideal fluids. The key idea is to regard the group of orientation-preserving diffeomorphism of the Euclidean space (corresponding to spatial configurations of an ideal incompressible fluid permeating the whole space) as a multisymplectic action on R^3 with the standard volume form seen as a 2-plectic form.

As a first result, we can explicitly construct a homotopy co-momentum map (a la Callies, Fregier, Rogers and Zambon) associated to this multisymplectic action showing that it correctly transgresses to the standard hydrodynamical co-momentum map defined by Arnol'd, Marsden and Weinstein and others.

The transition to knots occurs when one considers vortex filaments in hydrodynamics. It is possible to associate to these peculiar configurations of the fluid suitable conserved quantities, as defined by Ryvkin, Wurzbacher and Zambon. These quantities are directly related to the Gauss linking number of the link supporting the vorticity.

Time permitting, we shall discuss a reinterpretation of the (Massey) higher order linking numbers in terms of conserved quantities within the multisymplectic framework, giving rise to knot theoretic analogues of first integrals in involution.

A connection between these two topics can be established via mechanics of ideal fluids. The key idea is to regard the group of orientation-preserving diffeomorphism of the Euclidean space (corresponding to spatial configurations of an ideal incompressible fluid permeating the whole space) as a multisymplectic action on R^3 with the standard volume form seen as a 2-plectic form.

As a first result, we can explicitly construct a homotopy co-momentum map (a la Callies, Fregier, Rogers and Zambon) associated to this multisymplectic action showing that it correctly transgresses to the standard hydrodynamical co-momentum map defined by Arnol'd, Marsden and Weinstein and others.

The transition to knots occurs when one considers vortex filaments in hydrodynamics. It is possible to associate to these peculiar configurations of the fluid suitable conserved quantities, as defined by Ryvkin, Wurzbacher and Zambon. These quantities are directly related to the Gauss linking number of the link supporting the vorticity.

Time permitting, we shall discuss a reinterpretation of the (Massey) higher order linking numbers in terms of conserved quantities within the multisymplectic framework, giving rise to knot theoretic analogues of first integrals in involution.

Reduction plays an important role in both classical and quantum mechanics.
Using situations from Poisson geometry and deformation quantization as guiding examples, we define so called coisotropic algebras as algebraic abstractions of certain reduction schemes. Since one is always interested in representations of algebraic objects we construct a bicategory of coisotropic algebras and bimodules that allows us to compare coisotropic algebras by means of Morita equivalence. Finally, we show that reduction is well-behaved with respect to Morita equivalence and taking the classical limit.

Lecture 4

Depending on how the mini-course progresses this lecture would ideally contain the work in progress to develop a derived geometry/homotopical formalism for geometry of PDEs. Relations to synthetic geometry of PDEs of Khavkhine and Schreiber and the HAC for differential operators of Poncin will also be given.

Depending on how the mini-course progresses this lecture would ideally contain the work in progress to develop a derived geometry/homotopical formalism for geometry of PDEs. Relations to synthetic geometry of PDEs of Khavkhine and Schreiber and the HAC for differential operators of Poncin will also be given.

Lecture 3

The goal of this lecture will be to summarize what the point of and what was achieved in PTVV, CPTVV. Basically we will dive into all things `shifted-symplectic'.

Hour 1: Shifted Symplectic Structures in DAG, what was so interesting about the works of PTVV and CPTVV.

Hour 2: Three different approaches to shifted symplectic things in derived differential geometry. These will be coming from the examples of DDG?s that I survey in hour 2 of lecture 2.

The goal of this lecture will be to summarize what the point of and what was achieved in PTVV, CPTVV. Basically we will dive into all things `shifted-symplectic'.

Hour 1: Shifted Symplectic Structures in DAG, what was so interesting about the works of PTVV and CPTVV.

Hour 2: Three different approaches to shifted symplectic things in derived differential geometry. These will be coming from the examples of DDG?s that I survey in hour 2 of lecture 2.

Lecture 2

The goal of this lecture will be to provide some introduction to higher categorical mathematics (∞-categories and model categories, specifically 2-categories and categories of fibrant objects), and to introduce and focus on some DDG's which use these languages.

Hour 1: Explain the essence of an ∞-category, model categories, category of fibrant objects, extended example of 2-category.

Hour 2: Survey some DDG?s (probably those of D. Joyce in relation to Borisov-Noel and Spivak and Carchedi-Roytenberg, J. Pridham, and K. Costello, R. Grady, O. Gwilliam)

The goal of this lecture will be to provide some introduction to higher categorical mathematics (∞-categories and model categories, specifically 2-categories and categories of fibrant objects), and to introduce and focus on some DDG's which use these languages.

Hour 1: Explain the essence of an ∞-category, model categories, category of fibrant objects, extended example of 2-category.

Hour 2: Survey some DDG?s (probably those of D. Joyce in relation to Borisov-Noel and Spivak and Carchedi-Roytenberg, J. Pridham, and K. Costello, R. Grady, O. Gwilliam)

Derived Algebraic Geometry (DAG) is a well established area of modern mathematics, whose origins lie in the seminal works of Jacob Lurie and Toen-Vezzosi, but deciphering what actually constitutes a derived geometry, from the point of view of a `working mathematician', is not an easy task. The difficulties which arise predominantly emerge from the fact that the literature has an impressive reputation for its impenetrable nature, both with respect to the size (ie. Lurie's DAG - 900 pages) and its use of heavy ∞-categorical technologies. Not only this, but its foundations lie in algebraic geometry which makes it a little inaccessible to differential geometers and mathematical physicists who wish to make use of its powerful techniques.

In DAG one often defines the notion of derived scheme using the functor of points approach. Then, to give a derived stack over a field of characteristic zero is to give a functor from the category of commutative differential graded algebras to the category of simplicial sets, satisfying appropriate (homotopy) sheaf conditions. Actually, in all of its glory, what may be called derived E∞-geometry, is a most comprehensive version of DAG where the spaces are locally modelled on E∞-rings, as opposed to simplicial commutative rings or dg-algebras. In this version, one is implementing a theory of higher geometry in the (∞,1)-topos over the (∞, 1)-site of formal duals of E∞ rings, equipped with the etale topology.

In this mini course we will explain what it means to be a `derived space', why people are interested in derived geometry (both algebraic and differential) and why higher sheaf and category theory, specifically the theory of higher stacks, are ubiquitous in this derived world. We will talk about the various `relaxations' of the notion of a derived space, resulting in a journey from ∞-categorical setting to a formalism which is much more tangible, with our focus on explaining and motivating why these objects are used rather than proving technical results. We will survey the various derived theories out there and explain some pros and cons of each, depending on what one wants to achieve. We will focus on higher stacks and their derived version, with particular emphasis on Deligne-Mumford stacks. Additionally we will discuss and compare various theories of derived smooth stacks and we will talk briefly about L∞ spaces, derived Lie algebroids in differential geometry and derived foliations in algebraic geometry, as well as the famous shifted symplectic structures of PTVV,CPTVV.

Time permitting, we will discuss an ongoing work developing derived geometry of PDEs and how a systematic study of the derived symplectic geometry of (generally very singular) moduli spaces of Euler-Lagrange equations in QFTs, can be facilitated by this language of derived smooth geometry.

Lecture 1

The goal of this lecture will be to clarify and give meaning to the word derived in the general context of geometry, whether algebraic, differential, arithmetic or analytic. Some history of algebraic geometry leading to the birth of DAG will be given with a particularly heavy emphasis on the philosophy behind it. We will also discuss what types of problems derived geometric techniques can help us with.

Hour 1: From Bezout?s Theorem to Serre?s Intersection Formula, Pathological Mathematics and the Ambiguity of Identification.

Hour 2: What constitutes a geometry, what constitutes a derived geometry, some discussion of what HAC means, what DAG means, and the numerous DDGs out there.

End with a look towards Lecture 2, where we will see detailed examples of some DDGs.

In DAG one often defines the notion of derived scheme using the functor of points approach. Then, to give a derived stack over a field of characteristic zero is to give a functor from the category of commutative differential graded algebras to the category of simplicial sets, satisfying appropriate (homotopy) sheaf conditions. Actually, in all of its glory, what may be called derived E∞-geometry, is a most comprehensive version of DAG where the spaces are locally modelled on E∞-rings, as opposed to simplicial commutative rings or dg-algebras. In this version, one is implementing a theory of higher geometry in the (∞,1)-topos over the (∞, 1)-site of formal duals of E∞ rings, equipped with the etale topology.

In this mini course we will explain what it means to be a `derived space', why people are interested in derived geometry (both algebraic and differential) and why higher sheaf and category theory, specifically the theory of higher stacks, are ubiquitous in this derived world. We will talk about the various `relaxations' of the notion of a derived space, resulting in a journey from ∞-categorical setting to a formalism which is much more tangible, with our focus on explaining and motivating why these objects are used rather than proving technical results. We will survey the various derived theories out there and explain some pros and cons of each, depending on what one wants to achieve. We will focus on higher stacks and their derived version, with particular emphasis on Deligne-Mumford stacks. Additionally we will discuss and compare various theories of derived smooth stacks and we will talk briefly about L∞ spaces, derived Lie algebroids in differential geometry and derived foliations in algebraic geometry, as well as the famous shifted symplectic structures of PTVV,CPTVV.

Time permitting, we will discuss an ongoing work developing derived geometry of PDEs and how a systematic study of the derived symplectic geometry of (generally very singular) moduli spaces of Euler-Lagrange equations in QFTs, can be facilitated by this language of derived smooth geometry.

Lecture 1

The goal of this lecture will be to clarify and give meaning to the word derived in the general context of geometry, whether algebraic, differential, arithmetic or analytic. Some history of algebraic geometry leading to the birth of DAG will be given with a particularly heavy emphasis on the philosophy behind it. We will also discuss what types of problems derived geometric techniques can help us with.

Hour 1: From Bezout?s Theorem to Serre?s Intersection Formula, Pathological Mathematics and the Ambiguity of Identification.

Hour 2: What constitutes a geometry, what constitutes a derived geometry, some discussion of what HAC means, what DAG means, and the numerous DDGs out there.

End with a look towards Lecture 2, where we will see detailed examples of some DDGs.

In this talk, we discuss the notions of duality in Jacobi geometry. In the first part, based on joint work with A. Blaga, M. A. Salazar, and C. Vizman, we introduce the notion of a contact dual pair as a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. The central motivating example is formed by the source and the target maps of a contact groupoid. One of the main results is the characteristic leaf correspondence theorem for contact dual pairs which finds immediate application to the context of reduction theory. Indeed any free and proper contact groupoid action naturally gives rise to a contact dual pair and so the characteristic leaf correspondence yields a new insight into the global contact reduction as described by Zambon and Zhu. In the second part, based on joint work with J. Schnitzer, we discuss (weak) dual pairs in Dirac-Jacobi geometry. Our main result is an explicit construction of self-dual pairs for Dirac-Jacobi structures. As applications of this result we give a global and more conceptual construction of contact realizations and present a different approach to the normal form theorem around Dirac-Jacobi transversals.

The study of VB-groupoids, vector bundles in the category of Lie groupoids, naturally leads to the study of simplicial vector bundles, once we apply the nerve functor.
The algebraic counterpart of this homotopy theory is the study of the model category of cosimplicial modules over cosimplicial algebras. We give an account of the general study of model categories and possible generalisations of the Dold-Kan correspondance, which relates the study of this category to the homological theory of dg-modules over dg-algebras.

The aim of this seminar talk is to give a short introduction to Jacobi and Dirac-Jacobi Geometry
and provide an existence
proof of Normal Forms for Dirac-Jacobi bundles and Jacobi brackets using
recent techniques from Bursztyn, Lima and Meinrenken. As a first application,
we provide a conceptual proof of the splitting theorems of Jacobi pairs which were
first proposed by Dazord, Lichnerowicz and Marle.

Given a smooth manifold M it turns out that a lot properties may be encoded in terms of the differential graded Lie algebra structure of the (smooth) Hochschild cochain complex of the algebra of smooth functions on M, i.e. in the space of polydifferential operators. In fact this complex can be adapted in various ways including importantly an adaptation to Lie algebroids (this generalizes or restricts the available notions of differential operators). One major tool in the use of the complex of polydifferential operators is the fact that it is formal, i.e. it is quasi-isomorphic as a strong homotopy Lie algebra to its cohomology differential graded Lie algebra. In this talk I will outline a reconceptualization of various methods used to prove this formality using the local formality as input. This is done by introducing the notion of resolution of strong homotopy Lie algebras.

General deformation theory over a field of characteristic 0 has been formalized in terms of the Maurer-Cartan space of a strong homotopy Lie algebra (often a dgla). This was done through the works of many mathematicians, notably Drinfeld, Kontsevich-Soibelman, Lurie, Manetti, Pridham, Deligne, Hinich and Getzler. The last three showed how to construct an infinity groupoid (Kan complex) modeling the moduli space of a deformation problem given a strong homotopy Lie algebra. In this talk I will discuss the analogous case of constructing this moduli space in the case of a curved strong homotopy associative algebra. This is the first step in an attempt to generalize the work on deformation theory completely from characteristic 0 to arbitrary characteristic. As a specific example of such a deformation problem I will showcase the deformation theory of 1-morphisms over non-symmetric operads.

In this last seminar, we leave the abstract framework to delve into the computation of some actual derived functor.
We present two important examples: Ext and Tor, which are the derived functors of the Hom and the tensor product functors, respectively, in the category of R-modules.
A list of general properties of Ext and Tor will be followed by explicit computations in the case the ring R is nice enough.
Finally, again in the case of a reasonable ring, we will uncover the reasons for the names "Ext" and "Tor": the first one represents an obstruction for extensions of abelian groups to be split, while the second one represents an obstruction for modules to be flat (without torsion).

An additive functor between abelian categories which is exact only on one side, can be derived to repair the lack of exactness.
In the first part of the talk, we will introduce delta-functors as a means to describe general abelian (co)homological theories and exhibit a unifying framework to talk about derived functors.
Then we will give a construction of the left derived functors of a right exact additive functor F in an abelian category with enough projectives. They act on an object A by taking the cohomology of the image of a projective resolution of A through F.
The second part of the talk will be devoted to prove that (co)chain (co)homology in any abelian category forms a universal delta-functor.
Finally, as an application, we will apply this machinery to define sheaf cohomology as the collection of the right derived functors of the functor of global sections. A connection is made between this homological algebraic approach and the sheaf theoretic one via Godement resolutions, using a sort of generalized De-Rham Theorem. By the end of the talk, it will be perhaps clearer why "sheaf homology" is not talked about as much as sheaf cohomology.

According to Grothendieck's celebrated Tohoku paper, abelian categories form the right context in which to view homological algebra.
We will give the definition of an abelian category and present the first examples.
It turns out that if the abelian category at hand has enough "nicely-behaved" objects, there is a canonical way of "deriving" funtors which are only exact on one way.
The second part of the talk will be concerned with describing these objects and prove that in many categories of interest (sheaves, chain complexes...) a generic object can be resolved (better yet, "replaced") by one of this kind.

The Dold-Kan correspondence, in its most basic version, is an equivalence of categories between simplicial modules over a commutative ring R and non-negatively graded chain complexes of R-modules, that sends homotopy groups to homology groups and preserves (weak) homotopy equivalences. I will discuss this classical theorem, that builds a bridge between homotopical and homological algebra, giving some details about its proof. Then I will describe how it can be extended to simplicial R-algebras, where the equivalence holds only "up to homotopy".

This talk will be mainly devoted to the definition of simplicial homotopy groups. There is a very natural definition that follows closely the classical one in algebraic topology, but unfortunately it does not work for general simplicial sets: this motivates the introduction of Kan complexes. The construction will also require the discussion of some more basics of simplicial sets. Finally, I will show that simplicial homotopy groups share a lot of properties with standard homotopy groups, such as long exact sequences associated to fibrations.

In this talk I will introduce the notion of simplicial object in a category and define the basic operations we can perform on them. Several examples will be given, showing that many classical constructions, particularly from algebraic topology, can be very well understood in this general framework. Finally, I will briefly talk about the Dold-Kan correspondence, that will be discussed in more details in the next seminars.