Programme

Although there exist closed manifolds admitting effective actions of a large groups of symmetries, once we fix a closed manifold it is natural to expect that the collection of finite (or compact Lie) groups acting on it is limited in some sense. In this survey talk I will review both old and new results that materialize this expectation.

We will motivate the introduction of generalized geometry and discuss the most basic definitions.

We will survey the current understanding of homomorphisms between mapping class groups of surfaces. In particular, we will discuss the structure of such homomorphisms, often under various topological and algebraic constraints.

We develop a desingularization scheme for Poisson manifolds whose transverse structures are of compact semisimple type. The desingularized models emerge as E-symplectic manifolds—symplectic avatars of the original Poisson manifolds—that provide smooth symplectic representatives and uncover unexpected connections with cosymplectic geometry.
This is joint work with Ryszard Nest.

We expand upon the definition of (BBB)-branes on the Hitchin moduli stack following Franco–Hanson, and give examples of interest originating from S-duality and the recently emerging theory of relative Langlands duality. This is joint work with Eric Yen-Yoo Chen.

I will introduce a notion of variation of Hodge-Lefschetz structure for compact non-Kähler manifolds, which provides a generalization of the well-studied variations of polarised Hodge structure for projective varieties. A key ingredient for our construction is a “Bianchi identity” for hermitian metrics on the manifold, motivated by similar equations appearing in the string theory literature. Joint work with Raul Gonzalez Molina and Arpan Saha.

In differential geometry, flat spaces can be seen as the “simplest” objects. For instance, a flat Riemannian manifold is locally Euclidean not only smoothly, but geometrically. In the presence of skew-symmetric torsion, the situation seems a bit richer, in the sense that there are many models for flat spaces: compact simple Lie groups and the 7-sphere. In this talk, I will briefly describe the flat situation in generalized Riemannian geometry, a theory which contains and expands Riemannian geometry with skew-symmetric torsion.

This talk is a follow-up of the previous talk “Actions of large finite abelian groups on product of manifolds “. We will see how the discrete degree of symmetry can be used to obtain rigidity results for manifolds which have similar topological properties to tori, like closed aspherical locally homogeneous spaces or closed connected oriented manifolds admitting a non-zero degree map to a nilmanifold.

A central question in transformation group theory is to understand which finite groups act effectively on a given manifold. While significant progress has been made, with the Man-Su theorem and the Carlsson-Baumgarten theorem standing as two landmark results, a complete answer remains out of reach.
Whereas these classical results concern actions of specific groups, our study focuses on actions of increasingly large sequences of finite abelian groups on a fixed manifold. This asymptotic phenomena is captured with the discrete degree of symmetry, an invariant analogous to the classical toral degree of symmetry for finite group actions. We raise a natural question about how the the discrete degree of symmetry behaves under products of manifolds and we provide initial results towards answering it.

We will begin with some background and motivation, such as Weil and Mostow-Prasad rigidity, the inclusion of hyperbolic geometry in the projective setting, and what we mean by turnover ends. Then, we will briefly outline the proof of the main results, namely: i) hyperbolic turnover cusps are rigid or open up and become totally geodesic generalized cusps, and ii) after any projective deformation, a hyperbolic 3-orbifold with all its ends of turnover type remains properly convex. If time permits, we will review an example of application of these two results.

The purpose of this talk is to give a short survey on optimal isosystolic inequalities, and explain why this is actually quite relevant to  study them from the Finsler geometry point of view. Based on joint works with J.C. Álvarez Paiva & K. Tzanev in one hand, and with T. Gil Moreno de Mora in the other hand.

Confoliations, introduced by Eliashberg and Thurston in their pioneering 1989 work, offer a unified framework for studying foliations and contact structures in dimension three. However, in higher dimensions the theory has remained largely undeveloped, with no clear consensus even on appropriate definitions. In this talk, we propose meaningful higher-dimensional notions of confoliations and sketch some ideas on how to extend symplectic non-fillability results for contact structures to this broader setting. This is joint work with Fabio Gironella.

A double Poisson vertex algebra (DPVA) is the algebraic structure underlying a Hamiltonian classical field theory when the field variables are noncommutative. It was introduced by De Sole–Kac–Valeri (2015), generalizing Van den Bergh’s approach to Poisson geometry over associative algebras (2008). Motivated by this notion, we define double Courant–Dorfman algebras. We show that they are in bijective correspondence with DPVAs that are freely generated in degrees 0 and 1, and that they induce known geometric structures on representation spaces. As non-trivial examples, we construct noncommutative exact Courant algebroids. To prove that they satisfy the axioms of double Courant–Dorfman algebras, we uncover a noncommutative variant of the usual Cartan differential calculus, solving a problem posed by Crawley-Boevey–Etingof–Ginzburg (2007). Joint work with David Fernández (UPM) and Reimundo Heluani (IMPA).

Conifold transitions, as conjectured by Reid, appear as central in the study of the moduli of Calabi-Yau threefolds. One of their main features is that they lead outside the Kähler world, hence to obtain a proper understanding of the spaces arising from these transitions, we need to come up with notions of “canonical geometries” in a non-Kähler (but still Calabi-Yau) setting, the most famous of which are identified by the Hull-Strominger system, which also lays at the center of a geometrized version of Reid’s fantasy. In this talk I will discuss how, in the case of reversed conifold transitions, we have several good ingredients to work on the equations of the Hull-Strominger system, in particular by briefly presenting a joint work with Cristiano Spotti as well as an ongoing one with Mario Garcia Fernandez.

In this talk, I will present how feedforward neural networks can be viewed as a special class of quiver representations (“network quivers”). In this framework, a fixed neural network architecture determines a quiver, the parameters of the network define a representation, and inputs and outputs are incorporated via framing, so that networks become algebraic objects. Each evaluation of data then corresponds to a point in the moduli of framed quiver representations, providing a geometric space of the possible outputs of the network. This perspective offers new tools to analyze neural networks through geometry and topology to understand their expressive power better.

The well-known Milnor-Wood inequality gives a bound on the Toledo invariant of a representation of the fundamental group of a compact surface in a group of Hermitian type. While a lot is known regarding the counting of maximal Toledo components, and their role in higher Teichmueller theory, the non-maximal case remains elusive. In this talk, I will present a strategy to count the number of such non-maximal Toledo connected components. This is joint work in progress with Brian Collier and Jochen Heinloth, building on previous works joint with Olivier Biquard and Roberto Rubio, and Olivier Biquard, Brian Collier and Domingo Toledo.