July 17, 2024, 15h00: Sebastian Daza (Sala Riunioni)
G-Structures on Orbifolds
Orbifolds generalize manifolds by allowing the presence of singularities, encoded (locally) as the quotient of an euclidean space by a finite group of automorphisms.
Orbifolds are interesting on their own, although they appear naturally in many fields of mathematics such as algebraic geometry, classification problems in differential geometry, Lie groupoids among others.
A worthwhile approach to understand a geometric structure on a space is by studying its space of automorphisms.
The theory of \(G\)-structures exploits this idea.
We adapt this framework to the orbifold setting and show how to build a correspondence between geometric structures on an orbifold \(O\) and triples \((P, G,\theta)\), with \(P=\text{Fr}(O)\) a subbundle of the frame orbibundle, \(G=\text{GL}_n(\mathbb{R})\) a closed subgroup acting on \(P\) and \(\theta=\Omega^1(P,\mathbb{R}^n)\) a tensorial differential form.
Using this dictionary one can pass from differential geometric problems on \(O\) to \(G\)-equivariant differential geometric problems on \((P,G,\theta)\).
Furthermore, if the orbifold is effective, the frame orbibundle \(\text{Fr}(O)\) has a manifold structure. Therefore, the theory of \(G\)-structures allows us to do differential geometry on singular spaces, orbifolds, by working with \(G\)-equivariant differential geometry on manifolds, non-singular spaces.
June 28, 2024, 16h00: Fabricio Valencia (Sala Riunioni)
Closed Geodesics on Riemannian Stacks
This seminar aims at presenting an existence result for at least one closed geodesic of positive length on a separated Riemannian stack.
Our approach is inspired by the Lusternik-Schnirelmann program to address this problem in the classical case, as done for instance by Guruprasad-Haefliger
for orbifolds, so that we describe some of the properties of the stacky energy functional over stacky curves, but restricted to the case of regular separated Riemannian stacks.
Then, we apply the desingularization of separated Riemannian stacks due to Posthuma-Tang-Wang. This is based on joint work in progress with C. Ortiz and L. Vitagliano.
June 28, 2024, 16h00: Fabricio Valencia (Sala Riunioni)
Closed Geodesics on Riemannian Stacks
This seminar aims at presenting an existence result for at least one closed geodesic of positive length on a separated Riemannian stack.
Our approach is inspired by the Lusternik-Schnirelmann program to address this problem in the classical case, as done for instance by Guruprasad-Haefliger
for orbifolds, so that we describe some of the properties of the stacky energy functional over stacky curves, but restricted to the case of regular separated Riemannian stacks.
Then, we apply the desingularization of separated Riemannian stacks due to Posthuma-Tang-Wang. This is based on joint work in progress with C. Ortiz and L. Vitagliano.
June 28, 2024, 15h00: Stephane Geudens (Sala Riunioni)
Stability of Foliations
A foliation is a decomposition of a manifold into submanifolds that fit together nicely.
I will talk about the stability problem for foliations, which aims to find conditions under which a given foliation only admits trivial deformations.
I will first review a classical stability result due to Hamilton, which concerns Hausdorff foliations. Then I will discuss a stability result for Riemannian foliations,
obtained jointly with Florian Zeiser, which extends Hamilton's result.
April 15-18, 2024: Cornelia Vizman (Sala Riunioni)
Discrete Differential Geometry (Mini-course).
Course Program:
1. Simplicial complexes
2. Discrete curves and surfaces; curvatures; discrete Gauss-Bonnet theorem
3. Discrete differential calculus; the discrete Laplacian
4. Simplicial homology; persistent homology
October 3, 2023, 11h00: Jonas Schnitzer (Sala Riunioni)
Deformations of Lagrangian \(Q\)-submanifolds
\(\mathbb{N}\)-graded symplectic \(Q\)-manifolds encompass a lot of well-known mathematical structures, such as Poisson manifolds, Courant algebroids, etc.
Their Lagrangian \(Q\)-submanifolds are of special interest since they simultaneously generalize coisotropic submanifolds, Dirac structures and also serve as boundary conditions in AKSZ sigma models.
In this talk, we set up their deformation theory inside a symplectic \(Q\)-manifold via strong homotopy Lie algebras, which generalizes known results including the deformation theory of coisotropic submanifolds and Dirac structures.
This is a joint work with Miquel Cueca.
July 26, 2023, 11h30: Francesco Cattafi (Sala Riunioni)
Multiplicative frame bundle of a Lie groupoid
It is well known that the collection of linear frames of a smooth \(n\)-manifold \(M\) defines a principal \(\mathrm{GL}(n,\mathbb{R})\)-bundle over \(M\) (called the frame bundle); more generally, this makes sense for any vector bundle.
Conversely, any principal bundle together with a representation induces an associated vector bundle; these processes establish therefore a correspondence between vector bundles on one side, and principal bundles with representations on the other side.
If instead of a manifold \(M\) we begin with a Lie groupoid \(\mathcal{G} \rightrightarrows M\), one can consider both the frame bundles of \(\mathcal{G}\) and of \(M\) and try to "close" the resulting diagram in a natural way.
The frame bundle of \(\mathcal{G}\) is however too big to support a Lie groupoid structure over the frame bundle of \(M\).
In this talk, I will discuss how to fix this issue by introducing a special class of frames which interact nicely with the groupoid structure ("multiplicative frames").
At the end, I will sketch how to generalise this construction to a correspondence between VB-groupoids (groupoid objects in the category of vector bundles) and PB-groupoids (groupoid objects in the category of principal bundles).
This is a work in progress based on on-going discussions/collaborations with Chenchang Zhu and Alfonso Garmendia.
June 29, 2023, 11h00: Bjarne Kosmeijer (Sala Riunioni)
Lie groupoids, convolution algebras and non-commutative geometry
For many applications in mathematical physics, it is important to get a grasp on 'equivariant theory' of Lie groupoids (generally interpreted in terms of the classifying space).
For group actions this is very well understood in terms of the 'non-commutative geometry' of the associated convolution algebra, opening the door for instance to using techniques from algebraic index theory to equivariant index theory for group actions.
Our aim is to generalize this programme to Lie groupoids.
In this talk we'll discuss recent work in this direction, relating the deformation theory of a Lie groupoid with the deformation theory of its underlying convolution algebra by ways of a chain map between the deformation complex of the groupoid and the Hochschild complex of the convolution algebra.
We will focus on two applications in the realm of geometry of this correspondence: in the first part we will discuss the relation with the van Est-map by seeing it as a classical limit of our chain map, in the second part we will discuss the relations with the adjoint representation up to homotopy, the search for a Lie bracket on the deformation complex and the notion of a multiplicative differential operator.
Most of the work is joint with Hessel Posthuma.
Geometric Killing Vector Fields on Riemannian Stacks
We give an infinitesimal description of an isometric Lie 2-group action. We define an
algebra of transversal infinitesimal isometries associated to any Riemannian n-metric on
a Lie groupoid which in turn gives rise to a notion of geometric Killing vector field on
a quotient Riemannian stack. In particular, if our Riemannian stack is separated then
we explain why the algebra formed by such geometric Killing vector fields is always finite
dimensional.
May 8, 2023, 10h00: Ilias Ermeidis (Sala Riunioni)
Deformations of Ideals in Lie Algebras
The main aim of this talk is to give a gentle introduction to deformation theory, focusing on the context of Lie algebras. Furthermore, we will cover, at least partially, the knowledge gap about deformations of ideals and we will see how double structures, in the sense of Mackenzie, are hidden in this problem.
On the size of the tangency set of a submanifold with respect to a distribution
These lectures are devoted to the size of the tangency set of a submanifold with
respect to a certain distribution on the ambiental Euclidean space, in terms of
the Hausdorff dimension. Estimates for the Hausdorff dimension of the tangency set
will be provided. We will pay some special attention to the tangency sets with
respect to some highly noninvolutive distributions, such as those spanned by
global linearly independent vector fields satisfying the Hormander condition.
April 27, 2023, 14h30: Fabricio Valencia (Sala Riunioni)
Isometric Lie 2-Group Actions on Riemannian Lie Groupoids
We define isometric actions of Lie 2-groups on Riemannian groupoids and exhibit some of their immediate properties and implications.
We mention an existence result which allows both to obtain 2-equivariant versions of the Slice Theorem and the Equivariant Tubular
Neighborhood Theorem and to construct bi-invariant groupoid metrics on compact Lie
2-groups.
We provide natural examples, transfer some classical constructions and explain how this notion of isometric 2-action yields a way to develop a 2-equivariant Morse theory on Lie groupoids.
April 17, 2023, 15h00: Antonio Maglio (P/19)
Shifted Contact Structures on Differentiable Stacks
In this talk we introduce 0-shifted and +1-shifted contact structures on differentiable
stacks, thus laying the foundations of shifted Contact Geometry. As a byproduct we show that
the kernel of a multiplicative 1-form on a Lie groupoid (might not exist as a vector bundle
in the category of Lie groupoids but it) always exists as a vector bundle in the category of
differentiable stacks, and it is naturally equipped with a stacky version of the curvature. Our
shifted contact structures are related to shifted symplectic forms via a Symplectic-to-Contact
Dictionary.
April 13, 2023, 15h00: Fabricio Valencia (Sala Riunioni)
Morse Theory on Lie Groupoids and their Differentiable Stacks
We adapt the Morse theory results we obtained for Lie groupoid to the setting of differentiable
stacks. As an interesting consequence we get Morse-like inequalities for the orbit
spaces of a separated differentiable stack.
We describe some features of the Morse–Smale dynamics in the Lie groupoid context. In
particular, motivated by Austin–Braam’s construction, we also sketch how to construct a
double cochain complex whose total cohomology allows us to recover the Bott-Shulman-
Stasheff cohomology of the underlying Lie groupoid. It is worth mentioning that our
double cochain complex also has a 2-equivariant version.
April 4, 2023, 11h00: Fabricio Valencia (Sala Riunioni)
Morse Lie Groupoid Morphisms
We define Morse Lie groupoid morphisms and specify their properties. We add into the
picture a Riemannian groupoid metric and describe the natural groupoid structure of both
the disk and sphere bundles over a nondegenerate critical subgroupoid. We mention some
interesting examples.
Further, we present our Lie groupoid version of the Morse Lemma. We define the level subgroupoids
of a Morse Lie groupoid morphism and describe their topological behavior whether or not
we cross by a nondegenerate critical orbit
March 27, 2023, 15h00: Fabricio Valencia (Sala Riunioni)
Riemannian Lie Groupoids
We present the notion of Riemannian groupoid which was introduced by del Hoyo and
Fernandes. Such a notion is one of the main ingredients we need to start describing
some features related to Morse theory over both Lie groupoids and stacks. We exhibit
some of the immediate implications derived from having a Riemannian groupoid metric
and mention some examples as well as results.
March 20, 2023, 15h00: Fabricio Valencia (Sala Riunioni)
Classical Morse-Bott Theory
We speak about the classical results of Morse theory and introduce the notion of
Morse–Bott function which will be of vital importance for us later on. We sketch
the construction of the Austin–Braam’s cochain complex which allows to recover the de
Rham cohomology of a compact and oriented manifold equipped with Morse–Bott function
verifying a Smale transversality condition. It is important to say that such a construction
also naturally enables us to recover the Cartan model for the equivariant cohomology
associated to an action by a compact Lie group.
March 16, 2023, 15h00: Fabricio Valencia (Sala Riunioni)
Some Aspects of Riemannian Submersions
The aim of this series of seminars is to present an approach to extend the results of classical
Morse theory to the context of Lie groupoids and their differentiable stacks. Some of the essential
results to be treated throughout the talks can be found in separated joint works with S. Herrera–Carmona and with C. Ortiz.
In this first seminar we describe the problems we are mainly interested in and introduce some basics on Riemannian
submersions.
September 25, 2019, 10h00: Ryszard Nest (Sala Riunioni)
Around Formality and Non-Commutative Calculus.
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.
September 11, 2019, 10h30: Alfonso Garmendia (Room F/6)
Quotients on the Holonomy Groupoid of a Singular Foliation.
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.
September 10, 2019, 10h00: Alfonso Garmendia (Sala Riunioni)
The Holonomy Groupoid and Morita Equivalence for Singular Foliations
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.
June 6, 2019, 10h30: Luca Vitagliano (Sala Riunioni)
Homogeneous G-structures
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.
Some Topics in Deformation Theory and Differential Graded Lie Algebras (Mini-course).
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.
March 26, 2019, 14h30: João Nuno Mestre (Room P/18)
Deformations of Symplectic Groupoids
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.
Multisymplectic Aspects of Link Invariants
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 \(\mathbb{R}^3\) 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.
March 18, 2019, 14h30: Marvin Dippell (Sala Riunioni)
Reduction and Morita Theory for Coisotropic Triples
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.
February 27, 2019, 14h30: Jacob Kryczka (Room F/6)
Derived Geometries IV
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.
February 21, 2019, 14h30: Jacob Kryczka (Room F/6)
Derived Geometries III
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.
February 14, 2019, 14h30: Jacob Kryczka (Room F/6)
Derived Geometries II
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)
February 7, 2019, 14h30: Jacob Kryczka (Room F/6)
Derived Geometries I
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.
Dual Pairs in Jacobi Geometry
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.
November 28, 2018: Stefano Palessandro (Sala Riunioni)
The Model Category of Cosimplicial Modules
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.
November 13, 2018: Jonas Schnitzer (Sala Riunioni)
Normal Forms in Jacobi Geometry
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.
October 25, 2018: Niek de Kleijn (Sala Riunioni)
Resolutions of Strong Homotopy Lie Algebras
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.
October 23, 2018: Niek de Kleijn (Sala Riunioni)
Lie Theory for Complete Curved Strong Homotopy Associative 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.
July 21, 2023, 11h00: Generoso Martusciello (P/19)
Il Doppio Complesso di Čech-de Rham
Dopo i lavori di Poincaré, sempre più importanza hanno assunto in Geometria le tecniche (co)omologiche.
In Geometria Differenziale lo strumento principale è la Coomologia di de Rham e, il Teorema di de Rham ne esprime, sorprendentemente, l'invarianza topologica.
In questo seminario, verrà presentata una dimostrazione del Teorema di de Rham che passa per la Coomologia di Čech di uno spazio topologico a valori in un (pre)fascio.
Nella prima parte del seminario, verranno richiamati i concetti fondamentali di algebra omologica: complessi di cocatene, mappe di cocatene, coomologia, omotopie algebriche, ecc.
Si proseguirà poi con l'apparato tecnico necessario alla dimostrazione del Teorema: complessi doppi (aumentati), complesso doppio (aumentato) di Mayer-Vietoris relativo ad un ricoprimento, complesso di Čech relativo ad un ricoprimento, (pre)fasci su uno spazio topologico, coomologia di Čech di uno spazio topologico a valori in un (pre)fascio.
Il seminario si concluderà con la dimostrazione del Teorema di de Rham.
May 16, 2019, 14h30: Stefano Palessandro (Sala Riunioni)
Stefano Palessandro: Derived Functors III
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).
March 27, 2019, 14h30: Stefano Palessandro (Sala Riunioni)
Stefano Palessandro: Derived Functors II
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.
March 20, 2019, 14h30: Stefano Palessandro (Sala Riunioni)
Stefano Palessandro: Derived Functors I
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.
January 16, 2019, 14h30: Pier Paolo La Pastina (Sala Riunioni)
Pier Paolo La Pastina: Simplicial Methods in Homological Algebra III
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".
December 4, 2018, 10h30: Pier Paolo La Pastina (Sala Riunioni)
Pier Paolo La Pastina: Simplicial Methods in Homological Algebra II
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.
November 21, 2018, 14h30: Pier Paolo La Pastina (Sala Riunioni)
Pier Paolo La Pastina: Simplicial Methods in Homological Algebra I
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.