Research interests

Here are the links to my profiles on the most common research platforms, listed in a roughly descending order of importance (to me) and of the care I devote to maintaining them:

• Arxiv: cattafi_f_1

• Orcid: 0000-0001-9785-0770

• Zentralblatt MATH: cattafi.francesco

• Google scholar: tGTddzMAAAAJ

• MathSciNet MR: 1177482

• Researchgate: Francesco-Cattafi

• Web of Science: AAC-5214-2020

• Scopus: 57191194765

Lie groupoids are multifaceted mathematical objects that represent local continuous symmetries. In the past few decades they have risen to the forefront of research, fueled by advances in Poisson geometry and mathematical physics, and they have in turn led to countless applications in areas such as noncommutative geometry, pseudodifferential operators, simplicial manifolds, derived geometry, PDEs, and topological field theories. Lie groupoids provide a unifying framework for various classical concepts, including not only Lie groups, but also Lie group actions, submersions, principal bundles and foliations. Furthermore, Lie groupoids are the global counterpart of Lie algebroids, the many-object generalisation of Lie algebras, which proved to be very useful in a range of research areas in mathematics as well as in modern physics.

Its origin and historical development reveal that Lie theory has always been deeply intertwined with geometric structures. In particular, many authors developed the unifying theory of G-structures, which includes a wide variety of geometric structures, such as Riemannian metrics, (almost) symplectic structures and (almost) complex structures. All these structures, and various problems related to them, can be handled using Lie groups and principal bundles, or, equivalently, via transitive Lie pseudogroups. In order to get rid of this transitivity hypothesis, we produced a new framework for geometric structures defined by a Lie pseudogroup, revisiting the pioneering ideas of Sophus Lie and Élie Cartan on Lie pseudogroups and geometric structures from a modern perspective. More concretely, we provided a theory of prolongations, and proved various kinds of integrability theorems. This framework includes G-structures and other classical formalisms, but extends as well to the "almost versions" of Γ-structures, which so far had never been treated for arbitrary Lie pseudogroups. The relevant objects which make this approach work are Lie groupoids endowed with a multiplicative "PDE-structure”, their actions and the related Morita theory. Poisson geometry provides the guiding principle to understand those objects, which are directly inspired from, respectively, symplectic groupoids, Hamiltonian actions and symplectic Morita equivalence.

I have also investigated how Lie groupoids provide a clearer framework to understand Cartan geometries. These objects stem from the famous Erlangen program by Felix Klein, who stated that every (possibly non-Euclidean) "geometry" should be described by a group of transformations; this led Élie Cartan to take such geometries as "standard model" and define his espaces généralisés. In modern language, Cartan geometries can be formalised in terms of principal bundles and vector valued 1-forms, called Cartan connections. I have introduced the more general notion of a Cartan bundle, which encompasses both Cartan geometries and G-structures as particular cases. Furthermore, I proved that the classical 1-1 correspondence between principal bundles and transitive Lie groupoids restricts to a correspondence between Cartan bundles and transitive Pfaffian groupoids. This allows one to study Cartan connections as multiplicative forms on the associated (transitive) Lie groupoid.

Adopting this approach we also obtained an application on the integrability of contact structures. Indeed, contact structures are, intuitively, "flat", due to the existence of Darboux charts, but they have an underlying G-structure which is not flat (its integrable version would be a symplectic foliation). In order to make this observation rigorous, Albert and Molino proposed in 1984 a different definition of integrability, "twisted" by the Heisenberg Lie algebra. While this point of view seems quite ad hoc, these integrability phenomena can be understood from the broad perspective of transitive pseudogroups. Indeed, the pseudogroup of symmetries of the trivial G-structure on R^n is transitive because it contains translations (i.e. transformations by the action of the abelian group (R^n,+), while the pseudogroup of symmetries of the canonical contact structure on R^n is transitive because it contains transformations by the action of the Heisenberg group. In order to clarify the role played by these two Lie groups (the abelian and the Heisenberg one), we had to zoom out and adopt the framework of Cartan bundles. We developed therefore a theory of Cartan type-extensions (Lie algebras extending in a suitable way the coefficients of the 1-form of a Cartan bundle) and of integrability with respect to them. The integrability with respect to, respectively, the abelian or the Heisenberg Lie algebra, yields precisely the integrability conditions for symplectic foliations and for contact structures. Our approach provided also more insights on the integrability of Cartan geometries and of higher order G-structures.

These groupoid-oriented investigations of Cartan connections led me also to the study of contact subriemannian manifolds, i.e. manifolds with a contact distribution and a Riemannian metric defined on it. An important question in this area, arising e.g. from control theory, aims at constructing invariants of such structures via Cartan connection-like objects, analogously to obtaining the ordinary curvature of a Riemannian manifold via its canonical Levi-Civita connection. However, the existing approaches always assume further hypotheses, e.g. asking the manifold to have constant symbol algebras, and in general they rely on Lie groups to model global symmetries. In order to tackle this problem from a more general point of view, one needs to consider point-dependent symmetries, which are modelled by Lie groupoids; the case of transitive groupoids recovers the known cases solved by Lie groups. More precisely, we are currently considering the (non-transitive) groupoid of subriemannian isometries of graded tangent spaces associated to any arbitrary contact subriemannian structure, and its natural "filtered differential forms". Furthermore, we are working on the construction of the non-transitive analogue of a Cartan connection, in order to extract invariants of the contact subriemannian structure directly from its curvature tensor.

A research direction which has naturally emerged in various contexts in Poisson geometry (starting with Poisson-Lie groups) focuses on multiplicative structures, i.e. geometric structures on the space of arrows of a Lie groupoid that satisfy suitable compatibility conditions with the groupoid multiplication. This perspective extends naturally to the infinitesimal level, exploring how geometric structures on the Lie groupoid "differentiate" to analogous structures on its Lie algebroid, called infinitesimally multiplicative. Relevant examples include multiplicative symplectic structures (symplectic groupoids), which integrate Poisson structures, and multiplicative Poisson structures (Poisson groupoids), which integrate Lie bialgebroids. The quest for multiplicative structures on Lie groupoids is strictly related to (and often motivated by) the description of such geometric objects on differentiable stacks, which can be viewed as Morita equivalence classes of Lie groupoids.

We have introduced the theory of multiplicative "PDE-structures" (Pfaffian groupoids); these are the objects underlying the jet bundle of a Lie pseudogroup. Their infinitesimal counterpart (Pfaffian algebroids) can be regarded as "linear PDE-structures" on a Lie algebroid, and generalise the classical Spencer operator on jet bundles. Both objects appear naturally in our work on Lie pseugroups and geometric structures, which led us also to the development of the concepts of Pfaffian Morita morphisms and Pfaffian Morita equivalences.

Another contribution to this programme on multiplicative structures involved the frame bundle of Lie groupoids. This is motivated by extending the classical correspondence between vector bundles and principal bundles to higher structures. Indeed, in differential geometry there are several natural instances where diagrams of Lie groupoids and vector bundles, together with suitable compatibilities, appear. They are known as vector bundle groupoids (VB-groupoids) and their theory has been fairly developed in the past decades, with applications e.g. to Poisson geometry, representations up to homotopy, deformation theory and non-commutative geometry. On the other hand, little was known about the principal bundle counterparts of these objects.

We have introduced a special class of frames of VB-groupoids which interact nicely with the groupoid structure, which defines a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid (generalising the general linear group). In turn, this led us to the general notion of a principal bundle groupoid (PB-groupoid) and to a correspondence between VB-groupoids and PB-groupoids. We are currently investigating its infinitesimal counterpart, i.e. a correspondence between VB-algebroids and PB-algebroids. Future related works involve a suitable notion of connection on these PB-objects and their interplay with Morita equivalence; the long-term goal will be the study of the frame bundle of a differentiable stack and of G-structures on stacks.

On a different direction, multiplicative regular foliations on Lie groupoids (foliated groupoids) and their infinitesimal counterparts (infinitesimal ideal systems on Lie algebroids) have also been studied. However, many foliations arising from geometric situations are not necessarily regular. Natural examples include orbits of a vector field, orbits of a Lie group action, or the natural partition into symplectic submanifolds of a Poisson manifold. These examples led to the introduction of various formalisms to tackle singular foliations, which allow the leaves to vary in dimension. However, foliations on Lie groupoids have so far been investigated only in the regular case.

We addressed this issue by exploiting Haefliger's approach to foliations, which involves an atlas of "foliation charts", related to each other by a cocycle with values in the pseudogroups of all diffeomorphisms. There are two main advantages of this approach. First, one can treat both regular and singular foliations at the same time: in the first case, the foliation charts are submersions, while in the second case they are just smooth maps (the singular leaves arise when the foliation charts jump in rank). Second, one can equip the foliation with a geometric structure transverse to the leaves: it is enough to ask that the cocycle takes values in a smaller pseudogroup of diffeomorphisms (for instance the pseudogroup of isometries, which leads to Riemannian foliations). We are currently working on transporting Haefliger's ideas to a multiplicative setting, i.e. develop a theory of multiplicative Haefliger structures on Lie groupoids, and apply it to relevant cases, e.g. that of the symplectic foliations on the space of arrows of a Poisson groupoid.

The formal theory of PDEs on manifolds studies systems of partial differential equations as intrinsic and coordinate-free geometric objects. The central notion is that of jet bundle, originally introduced by Ehresmann in the 50's, which provides a conceptual approach to the theory of prolongations, i.e. the procedure which associates to any PDE a new PDE with the same solutions but of a higher order, by adding its first-order differential consequences. In turn, one can investigate the formal integrability of PDEs on jet bundles by means of algebraic tools such as the Spencer cohomology, and identify the obstructions to the existence of solutions. Another powerful formalism in this area is that of exterior differential systems, which led to the solution of several classification problems in geometry and to a modern formulation of the Cartan-Kähler Theorem in the real analytic setting.

The fundamental information of a PDE is encoded in the Cartan form on jet bundles, a vector-valued differential 1-form used to detect solutions and symmetries. We have axiomatised the properties of the Cartan form in a way completely independent of the jet bundle where it originates. This point of view led us to define a general object, called a Pfaffian fibration, which can be thought of as a "PDE-structure" on a manifold. With this formalism we were able to provide a more conceptual approach to the theory of prolongations and of linearisation. This framework which constitutes a foundational step for our programme of revisiting Lie's and Cartan's works on Lie pseudogroups and geometric structures, motivated by the interpretation of Lie pseudogroups as pseudogroups generated by systems of PDEs.

Another important application of jet bundles involves the so-called variational sequences, i.e. complexes of (sheaves of) modules of differential forms on jet bundles. These sequences are instrumental to tackle problems originated from mechanics and classical field theory. In particular, one can detect important objects, such as the Lagrangian and the Euler-Lagrange equations, as elements of the sequence, as well as reformulate in this language fundamental results such as Noether’s theorems, which relate symmetries and conserved quantities.

We have focussed on Krupka’s variational sequence, proving a variational Cartan formula for (equivalence classes of) differential forms. Moreover, for any given Lagrangian, we determined the precise condition for a Noether–Bessel-Hagen current, i.e. an object associated to a generalised symmetry, to be variationally equivalent to a classical Noether current. We also showed that, if it exists, this Noether current is exact along the solutions of the Euler-Lagrange equations, and it generates a canonical conserved quantity.

Below are the MSC2020 classes most relevant to my present and past research, listed in a roughly descending order of current interests:

58H Pseudogroups, differentiable groupoids and general structures on manifolds

• 58H05 Pseudogroups and differentiable groupoids

53C Global differential geometry

• 53C05 Connections (general theory)

• 53C10 G-structures

• 53C12 Foliations (differential geometric aspects)

• 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

• 53C17 Sub-Riemannian geometry

58A General theory of differentiable manifolds

• 58A10 Differential forms in global analysis

• 58A15 Exterior differential systems (Cartan theory)

• 58A17 Pfaffian systems

• 58A20 Jets in global analysis

• 58A30 Vector distributions (subbundles of the tangent bundles)

53D Symplectic geometry, contact geometry

• 53D15 Almost contact and almost symplectic manifolds

• 53D17 Poisson manifolds; Poisson groupoids and algebroids

• 53D18 Generalized geometries (à la Hitchin)

• 53D20 Momentum maps; symplectic reduction

• 53D35 Global theory of symplectic and contact manifolds

57R Differential topology

• 57R25 Vector fields, frame fields in differential topology

• 57R30 Foliations in differential topology; geometric theory

• 57R57 Applications of global analysis to structures on manifolds

58J Partial differential equations on manifolds; differential operators

• 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

• 58J70 Invariance and symmetry properties for PDEs on manifolds

Miscellanea

• 22A22 Topological groupoids (including differentiable and Lie groupoids)

• 22E05 Local Lie groups

• 35A30 Geometric theory, characteristics, transformations in context of PDEs

• 35R01 PDEs on manifolds

• 37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)

70G General models, approaches, and methods in mechanics of particles and systems

• 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholo-

nomic, etc.) for problems in mechanics

• 70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics

70S Classical field theories

• 70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems

• 70S10 Symmetries and conservation laws in mechanics of particles and systems