SCHEDULE UPDATED WITH CHANGES

Below is a full agenda for the event. You can see more details such as titles/abstracts for the talks by clicking on each item. Please take note of the time zone information at the bottom. The events are scheduled in Lisbon time but, depending on your browser, you may see the events in your local time zone.

Abstracts

Monday

Marius Crainic: Poisson Manifolds of Compact Types - slides

In this talk, I’ll look back on a long-term collaboration with Rui Fernandes, centered around compactness questions in Poisson geometry, and developed together with David Martínez Torres. I’ll give an overview of what we’ve done so far. 

David Martínez Torres: Kahler Poisson structures and Hermitian symmetric spaces - slides

The Poincaré disk has a natural holomorphic embedding into the complex projective line. One verifies that the inverse of the Poicaré Kahler form extends to the projective line to a "(pseudo) Kahler" Poisson structure. It turns out that this property is common to any bounded symmetric domain. In the talk we shall discuss this interaction between Poisson geometry and Hermitian symmetric spaces.

This is ongoing joint work with P. Frejlich.

Nguyen Tien Zung: The tori, in math and in life - slides

Xiang Tang: Superconnection and Orbifold Chern Character - slides

The orbifold Chern character is a fundamental invariant for complex vector bundles on an orbifold. In this talk, we will extend this construction to coherent sheaves on complex orbifolds, following the ideas of Bismut, Block, Shen, and Wei on flat antiholomorphic superconnections. We will present a Riemann-Roch-Grothendieck theorem concerning the orbifold Chern character for orbifold embeddings. This result enables us to establish the uniqueness of the orbifold Chern character.

Matias del Hoyo: Simplicial Methods in Lie Theory - slides

There are good reasons why the theorems should be easy and the definitions hard. This talk is built around three hard definitions — and the theorems that come with them — developed over the past decade in the study of (higher) Lie groupoids from a simplicial viewpoint. First, I will survey the theory of Riemannian groupoids, developed with Rui Loja Fernandes: their role in the linearization problem, their various follow-ups, and some open questions that remain. Then, I will discuss higher vector bundles and representations up to homotopy, a higher Grothendieck correspondence recently established with Giorgio Trentinaglia. Finally, I will present ongoing work with Alejandro Cabrera, where we develop the higher Lie functor and establish Van Est-type theorems for higher Lie groupoids.

Tuesday

Eckhard Meinrenken: Van Est theory for double structures - slides

We construct a van Est homomorphism from the double complex of an LA-groupoid to the Weil algebra of the associated double Lie algebroid, and describe some of its properties. Based on joint work (in progress) with Maria Amelia Salazar.

María Amelia Salazar: Linearization around leaves of proper contact groupoids - slides

In this talk I will explain a local normal form around a symplectic leaf of a proper contact groupoid. Following the lines of the work of Crainic-Loja Fernandes-Martinez Torres in PMCT1, we construct a local linear model that captures the first jet of the Jacobi structure along the symplectic leaf, and prove that any leaf of a Jacobi manifold integrable by a proper contact groupoid has a tubular neighborhood that looks like the linear model. 

This is based on ongoing work with Camilo Angulo and Daniele Sepe.

Eugene Lerman: Some differential geometry over C∞-rings - slides

Recently Joyce, building on Dubuc's construction of C∞-schemes in 1980, developed foundations for algebraic geometry over C∞-rings. 

But what about differential geometry, i.e., things like vector fields and their flows, differential forms, Poisson brackets and Cartan calculus? I will describe my attempt to fill the gaps in literature.

Peter Olver: Convergence of Normal Forms of Submanifolds for Lie Pseudo-group Actions - slides

I will present a new theorem proving convergence of normal form series for a broad range of Lie pseudo-group actions on submanifolds.  This result generalizes the normal forms of Chern and Moser for hypersurfaces in CR geometry.  The proof relies on combining the theory of involutive systems of partial differential equations with the method of equivariant moving frames.  Joint work with Francis Valiquette and Masoud Sabzevari.

Jiang-Hua Lu: From log-canonical Poisson structures to cluster structures via Poisson deformations - slides

In this talk we show that the standard Poisson structure on a Schubert cell in the flag variety of a complex semi-simple Lie group is, in a sense, a master deformation of its log-canonical term, and we  explain a close relation between the Poisson cohomology classes appearing in the deformation and the mutation matrix for the standard cluster structure on the Schubert cell. The talk is partially based on joint work with M. Matviichuk.

Marco Gualtieri: Coping with incomplete double groupoids - slides

It is often said that Manin triples, which are fundamental in the study of Poisson Lie groups as well as generalized Kähler structures, are the infinitesimal objects corresponding to symplectic double groupoids. However, in many cases these double groupoids are incomplete: they fail to satisfy an important Kan condition which excludes them from being considered higher Lie groupoids. Focusing on the example of the generalized Kahler structures on compact Lie groups, we will see that the incomplete double groupoids involved actually sit within a larger Lie 2-groupoid, which itself integrates the Courant algebroid of the Manin triple we started with. Joint work in progress with Daniel Álvarez.

Wednesday

Pol Vanhaecke: An algebraically integrable recursion relation related to the Volterra lattice - slides

In a recent collaboration with Andy Hone and John Roberts, we studied a 4D birational map that appeared in a recent classification by Gubbiotti et al. of a specific type of 4D birational maps having two invariants. The main result is that this map is discrete algebraically integrable in a precise sense which I will explain in detail. It is related to the Volterra lattice, which I studied in 2000 (in the periodic case) with Rui.

Ieke Moerdijk: Operads, graphs and configuration spaces - slides

The notion of an E_n-operad is central in the theory of operads and its applications in topology, deformation theory, etc. However, the awkward situation is that there is no conceptual definition. Instead, there are only examples of many different kinds, and some confusing statements in the literature about the relation between these examples. We will present several of these examples, with an emphasis on operads of graphs and their relation to configuration spaces.

(reference: Beuckelmann-Moerdijk, A small catalogue of En-operads, ArXiv)

Ioan Mărcuț: Closed pseudogroups and their sheaf of Lie algebras - slides

The main goal of this talk is to introduce closed pseudogroups and their associated sheaf of Lie algebras. For this, the weak C-infinity topology Whitney on the sheaf of smooth sections of a fibre bundle will be discussed. This is used to define closed subsheaves, which generalize the sheaf of solutions to a PDE. The topology on the sheaf of smooth sections has certain particularities, which yields interesting characterizations of closed subsheaves. Finally, the main result of this talk it that closed pseudogroups admit a closed sheaf of Lie algebras, which could be seen as an analog of the closed-subgroup theorem of Cartan and van Neumann.

This work is part of a larger project, on rigidity of solutions to a PDE with symmetries, which was started in collaboration with Roy Wang. An early version of the results are contained in the PhD thesis of Roy Wang, available here: https://arxiv.org/abs/1712.00808.

Thursday

Daniele Sepe: Monotone vs Fano: an equivariant perspective - slides

Complex projective geometry does not just provide examples of compact symplectic manifolds, but also inspiration to define and study properties of symplectic forms. A case in point is the class of monotone symplectic manifolds: These are motivated by smooth Fano varieties and are those for which the symplectic form is cohomologically a positive multiple of the first Chern class. By the work of Reznikov and of Fano and Panov, not all monotone symplectic manifolds are Fano varieties; however, the two classes coincide in the presence of a toric action. The latter result indicates that the two classes should be related in the presence of "large" symmetries. The aim of this talk is to discuss this problem in the presence of a complexity one Hamiltonian torus action, i.e., one for which the reduced spaces are two-dimensional. This is joint work with Isabelle Charton and Silvia Sabatini.

Eva Miranda: A Geometric Route to Poisson Desingularization - slides

Resonating with geometric ideas developed by Rui Fernandes and Marius Crainic in their work on Conn’s linearisation theorem, we propose a method to desingularize Poisson structures that admit transverse Poisson structures of semisimple compact type. Our construction uses E-symplectic manifolds to model the desingularized geometry and reveals unexpected connections with cosymplectic structures.

This is joint work with Ryszard Nest.

Henrique Bursztyn: Poisson cohomology of certain Poisson structures on spheres - slides

For each $n\geq 2$, I will discuss a certain Poisson structure $\Lambda_n$ on the sphere $S^n$. For n=2, it is a log-symplectic "necklace" Poisson structure (previously studied by Roytenberg). For n=3, it is the Lu-Weinstein Poisson structure on $S^3=SU(2)$. I will discuss a general linearization result for these Poisson structures and use it to compute their Poisson cohomology. This is joint work with Hudson Lima.

Marco Zambon: Coisotropic branes in symplectic geometry - slides

A brane in a symplectic manifold M is a coisotropic submanifold N together with a closed 2-form which is compatible in a specific sense. Despite being defined in terms of symplectic geometry, branes involve complex geometry in an essential way. We will state some results and open questions in the case N=M (space-filling branes), i.e. the case in which M carries a holomorphic symplectic form. For branes supported on lower-dimensional submanifolds, we then address the question of infinitesimal deformation of branes, and whether all coisotropic submanifolds nearby a given one are themselves branes. This talk is based on ongoing work with Charlotte Kirchhoff-Lukat.

Anton Alekseev: Towards tropical Poisson-Lie groups - slides

According to Drinfeld, Poisson-Lie groups come in pairs: to a Poisson-Lie group G corresponds the dual Poisson-Lie group G^*, and there is a Poisson action (called the dressing action) of G on G^*.  The most studied examples of such pairs are due to Lu-Weinstein with G a compact Lie group with the standard Poisson structure \pi, and G^* a solvable Poisson-Lie dual with Poisson structure \pi^*. One can scale both Poisson structures with a real parameter s to obtain a pair (G, s\pi), (G^*, s\pi^*). The limit s --> 0 gives rise to the Hamiltonian action of G on the space Lie(G)^* endowed with the KKS Poisson structure.

In this talk, we will discuss various aspects of the « tropical » s --> \infty limit of the scaling parameter. We will focus on the example of G=U(n). In that case, in appropriate coordinates, the Poisson structure s\pi^* tends to a constant Poisson structure on a product of the Gelfand-Zeitlin cone and a torus. To discuss the action of G on G^ *, we use ideas from the Berenstein-Kazhdan theory of geometric crystals. While the action of the Lie algebra Lie(G) on G^* has no meaningful limit for s-->\infty, there is a well defined limit of the corresponding action Lie algebroid. We consider the example of n=2, and we speculate on the case of n>2.

The talk is based on a work in progress with A. Berenstein, A. Gurenkova, and Y. Li.

Friday

Camille Laurent-Gengoux: Singular leaves

I will explain in detail a work of my student Oscar Cosserat about how to use symplectic groupoids to construct explicit Poisson integrators that will never "get out" of a singular leaf, and compare it with its predecessors (and successors). I will then sketch a construction by Simon Raphael Fischer of a Yang-Mills bundle associated to a singular leaf of a singular foliation and explain why it implies that, at formal level, there are not-so-many possible transverse Poisson structures when the symplectic leaf and the transverse Poisson structure is given, and how to classify them. Time remaining, I will say a few words of a method of resolutions of singular leaves of Lie algebroids made by Ruben Louis following a construction by Omar Mohsen.

Ana Cannas da Silva: Toric Real Loci via Moment Polytopes - slides

Any symplectic toric manifold admits anti-symplectic involutions anti-commuting with the toric action. Toric real loci are the lagrangian submanifolds obtained as fixed point sets of such involutions. We use a particular polyhedral description — called a kaleidoscope — for a toric real locus, to understand, in an elementary geometric way, its orientability and Euler characteristic. The kaleidoscope definition relies on the restriction of the moment map to a toric real locus and provides a user-friendly and computationally simple description of that toric real locus. This is joint work with João Camarneiro.

Alejandro Cabrera: Quantization, integrability and going non-formal with Rui - slides

In this talk, the idea is to give an overview of our on-going project, together with Rui, on non-formal quantization and integrability. The key objective is to explain a novel phenomenon: when a Poisson manifold is not integrable (to a Lie groupoid), then this is translated into the failure of any non-formal star-product quantization to be fully associative. This phenomenon is lost upon asymptotic expansion to a formal star product and thus requires the specification of which kind of "non-formal" star products we take. We then explain the necessary concepts from semiclassical analysis and Fourier integral operators, outline the arguments behind this phenomenon and illustrate it with simple examples.

Miguel Abreu: Fano Bott manifolds and extremal Kahler metrics - slides

Fano Bott manifolds are particular examples of iterated $\CP^1$-bundles. They are also toric manifolds with reflexive moment polytopes that are combinatorically equivalent to the reflexive hypercube. In this talk I will present recent results of Higashitani-Kurimoto (2022) and Cho-Lee-Masuda-Park (2023, 2024) on their classification and discuss what is (un)known about their extremal Kahler metrics. This talk is motivated by an ongoing joint project with Maarten Mol and my good old friend Rui Loja Fernandes.