Abstracts 

Abstracts PDF  (Clickable titles lead to the slides)

Anton Alexeev

Batalin-Vilkovisky structures on moduli of flat connections.

Let G be a Lie group with an invariant scalar product on its Lie algebra, and S be a closed oriented 2-manifold. Then, by results of Atiyah-Bott and Goldman, the corresponding moduli space of flat connections M(S, G) carries a Poisson structure (a symplectic structure if S has no boundary).

We will show that if G is a unimodular Lie supergroup with an invariant odd scalar product on its Lie super algebra, then the moduli space of flat connections M(S, G) carries a Batalin-Vilkovisky (BV) structure. Our main tool is an odd counterpart of the Fock-Rosly formula which defines this canonical BV structure. It time permits, we will also outline some open problems.

The talk is based on a joint work with F. Naef, J. Pulmann, and P. Severa, see arXiv:2210.08944.

Alexei Kotov

DG manifolds, complex geometry and deformations

A DG manifold is a Z-graded supermanifold equipped with a homological vector field of degree 1. It is well known that DG manifolds play an important role in physics, providing the mathematical basis for the BV-BRST formalism. More precisely, such a manifold is used to describe the equations of motion and gauge symmetries simultaneously. In this talk I will focus on the interaction between DG manifolds and DG bundles (fiber bundles in the category of DG manifolds) and "classical" geometry. I will show examples of how DG manifolds and DG bundles encode information in the context of complex geometry and how they produce some natural structures in deformation theory.

Giuseppe Marmo

Hadamard's descend method: Electrodynamics.

In his lectures on Partial Differential Equations, Hadamard dealt with solutions of "cylindrical wave-equations" by means of solutions of "spherical wave-equations”.

If one tries to apply his method directly to Maxwell equations, one obtains wrong results. The method has to be improved because of the intervention of constitutive equations connecting Faraday field and Ampere’s field, the improved descend method requires not only a reduction of space-time but also a "chance-of-type" for involved fields. In the reduction procedure some components of the initial two-form give rise to one-forms and possibly to zero-forms. We shall present a coordinate-free procedure to implement Hadamard’s improved method.

The reduced equations turn out to be independent, and overall they suggest a possible way on how to unify independent equations on lower-dimensions by going to higher dimensions.

Juan Carlos Marrero

Poisson geometric integrators from retraction maps for integrable Lie algebroids.

A Lie algebroid structure on a vector bundle is equivalent to a graded Lie algebra structure on the space of skew-symmetric multisections of the vector bundle. Global objects associated with Lie algebroids are Lie groupoids. In this talk, I will present an extension of the notion of a retraction map for the Lie algebroid of a Lie groupoid. The resultant theory will be applied in obtaining geometric integrators for the reduction of Lagrangian and Hamiltonian systems which are invariant under the action of a symmetry Lie group. These integrators will preserve the induced Poisson structure on the reduced system. The results presented in this talk are part of a paper in collaboration with M Barbero and D Martin.

Manuel de León

Graded Poisson and graded Dirac structures.

There have been several attempts in recent years to extend the notions of symplectic and Poisson structures in order to create a suitable geometrical framework for classical field theories, trying to achieve a success similar to the use of these concepts in Hamiltonian mechanics. These notions always have a graded character, since the multisymplectic forms are of a higher degree than two. Another line of work has been to extend the concept of Dirac structures to these new scenarios. In the present paper we review all these notions, relate them and propose and study a generalization that (under some mild regularity conditions) includes them and is of graded nature. We expect this generalization to allow us to advance in the study of classical field theories, their integrability, reduction, numerical approximations and even their quantization.

Edith Padrón

Reduction of Time Dependent Hamiltonian Systems with Momentum Map.

In 1986, Albert proposed a reduction process for cosymplectic structures with momentum maps. In this talk, we analyze the limitations of that theory in the context of the reduction of symmetric time-dependent Hamiltonian systems. We will show that there are significant examples in this setting where Albert’s reduction doesn't work. Based on this analysis, we conclude that cosymplectic geometry is not suitable for this type of reduction. Motivated by this observation, we propose a new approach to perform reduction on such Hamiltonian systems, one that includes the examples for which Albert’s reduction fails.

Yunhe Sheng

Categorification of Courant algebroids and Symplectic NQ manifolds of degree 3.

We introduce the notion of a CLWX 2-algebroid and show that symplectic NQ manifold of degree 3 gives rise to a CLWX 2-algebroid. This is the higher analogue of the result that a QP-structure of degree 2 gives rise to a Courant algebroid. A CLWX 2-algebroid can also be viewed as a categoried Courant algebroid due to the fact that its section space is a Leibniz 2-algebra, which is the categorification of Leibniz algebras.

Vladimir Salnikov

Generalized geometry for modelling in mechanics.

In this talk I will describe some concepts from 'generalized geometry' that appear naturally in the qualitative analysis of mechanical systems. I will sketch an approach to variational formulation of dynamics on Dirac structures ([1-2]), and (try to) position it in the context of the results from 'geometric mechanics' of the last couple of decades. I will also present some recent developments of those results, related in particular to symplectic and Poisson reduction. Time permitting, I will mention some application of the approach to design of numerical methods and some extensions to infinite dimensional settings.

[1] V.Salnikov, A.Hamdouni, D.Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, 2021

[2] O. Cosserat, C. Laurent-Gengoux, A. Kotov, L. Ryvkin, V. Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023

Thomas Strobl

Gauge theories beyond groups and their underlying fiber bundles.

Standard gauge theories are based on principal G-bundles and associated bundles, where G is a (finite-dimensional) Lie group. However, there are gauge theories that do not fit within this framework, including the Poisson sigma model, gauged standard sigma models with a metric that does not admit isometries, and curved Yang-Mills-Higgs gauge theories. For this purpose we introduce principaloid bundles P, which are based on structure Lie groupoids. These are honest fiber bundles and as such fundamentally different from principal groupoid bundles. We show that ordinary principal bundles, their associated bundles, as well as general fiber bundles are particular examples. We study connections on them, their gauge transformations, and construct the Atiyah groupoid of P, which governs its symmetries.

Rudolf Šmolka

Flows of Graded Vector Fields

Similarly as for smooth manifolds, we can define vector fields on a Z-graded manifold algebraically - as derivations of its algebra of graded functions. However, unlike for smooth manifolds, we no longer have the luxury of a geometric interpretation in terms of integral curves. Therefore, if we would like to talk about flows of graded vector fields, we must make a different approach. In this talk, we outline one such approach and discuss some features of the so obtained graded flows. For example, we will see how the properties of a flow change based on the degree of the vector field.

Alfonso Giuseppe Tortorella

Deformations of Symplectic Foliations.

In this talk, based on joint work with Stephane Geudens and Marco Zambon, we develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result is that each symplectic foliation is attached with a cubic L∞ algebra controlling its deformation problem. Indeed, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer–Cartan elements of the associated L∞ algebra. Further, we prove that, under this one-to-one correspondence, the equivalence by isotopies of symplectic foliations agrees with the gauge equivalence of Maurer–Cartan elements. Finally, we show that the infinitesimal deformations of symplectic foliations can be obstructed. 

Luca Vitagliano

Homogeneous Boundaries of Geometric Structures.

Under appropriate homogeneity conditions, a hypersurface in a symplectic manifold inherits from the ambient a contact or a cosymplectic structure. There are similar statements for Poisson manifolds as well as for complex manifolds. Using ideas from the homogeneous symplectic approach to Contact Geometry, mainly promoted by Janusz Grabowski and his collaborators, we present a very general theorem putting all these statements under the same umbrella. This also allows generalizations, e.g., to Dirac Geometry, Generalized Complex Geometry and G-structures. This is joint work with Alfonso Tortorella.

Jan Vysoký

Introduction to Graded Vector Bundles.

The category of graded vector bundles over graded manifolds is introduced. Inspired by supergeometry and algebraic geometry, they are defined as certain sheaves of graded modules over the structure sheaf of the base graded manifold. 

In ordinary geometry, Serre-Swan theorem relates a geometrical definition of vector bundles to finitely generated projective modules. This fundamental result allows one to work with vector bundles in an entirely algebraic way. It is expected that graded vector bundles correspond to finitely generated projective modules in a similar fashion. However, since graded vector bundles cannot be described by their fibers, one cannot use the standard arguments to prove the theorem. A sketch of the proof is presented.

Ping Xu

BV∞ quantization of (-1)-shifted derived Poisson manifolds.

In this talk, we will give an overview of (-1)-shifted derived Poisson manifolds in the  C∞ context, and discuss the quantization problem. We describe the obstruction theory and prove that the linear (-1)-shifted derived Poisson manifold associated to  any L∞-algebroid admits a canonical  BV∞ quantization. This is a  joint work with Kai Behrend and Matt Peddie.

Alessandro Zampini

From 3d Poisson manifolds to 4d non-commutative manifolds.

The idea of the talk is to describe a path whose origins trace back to several paper co-authored by Janusz in the ‘90s on three dimensional Poisson manifolds, and ends  upon  defining  a set of non-commutative four dimensional manifolds.