**Conference organized on the occasion of Simone Gutt’s 65th birthday**

IES Cargèse

October 31 – November 4, 2022

## Speakers:

- Mélanie Bertelson (Université libre de Bruxelles, BE)
- Pierre Bieliavsky (Université Catholique de Louvain, BE)
- Martin Bordemann (Université de Haute Alsace, FR)
- Frédéric Bourgeois (Université Paris-Saclay, FR)
- Henrique Bursztyn (Instituto de Matemática Pura e Aplicada, BR)
- Alberto Cattaneo (Universität Zürich, CH)
- Ezra Getzler (Northwestern University, US)
- Joel Fine (Université Libre de Bruxelles, BE)
- Jean Gutt (Institut National Universitaire Champollion, FR)
- Sonya Hohloch (Universiteit Antwerpen, BE)
- Niky Kamran (Université McGill, CA)
- Laurent La Fuente Gravy (Université libre de Bruxelles, BE)
- Camille Laurent-Gengoux (Université de Lorraine, FR)
- Eva Miranda (Universitat Politècnica de Catalunya, ES)
- Sylvie Paycha (Universität Postdam, DE)
- Martin Schlichenmaier (Université du Luxembourg, LU)
- Wilfried Schmid (Harvard University, US)
- Lorenz Schwachhöfer (Universität Dortmund, DE)
- Stefan Waldmann (Universität Würzburg, DE)
- Alan Weinstein (University of California at Berkeley and Stanford University, US)
- Marco Zambon (Katholieke Universiteit Leuven, BE)

## Scientific Committee :

Mélanie Bertelson, Michel Cahen, John Rawnsley, Stefan Waldmann, Alan Weinstein

## Organizing Committee :

Mélanie Bertelson, Michel Cahen

# Program :

Monday | Tuesday | Wednesday | Thursday | Friday | |

9:00 | Jean Gutt | Mélanie Bertelson | Martin Bordemann | Laurent Lafuente-Gravy | Eva Miranda |

10:00 | coffee break | coffee break | coffee break | coffee break | coffee break |

10:15 | Pierre Bieliavsky | Stefan Waldmann | Frédéric Bourgeois | Sonja Hohloch | Sylvie Paycha |

11:15 | Ezra Getzler | Joel Fine | Marco Zambon | Henrique Bursztyn | Camille Laurent-Gengoux |

12:15 | lunch – IESC | lunch – IESC | lunch – IESC | lunch – IESC | lunch – IESC |

13:30 | discussions | discussions | discussions | discussions | Alberto Cattaneo |

15:00 | coffee break | coffee break | coffee break | coffee break | coffee break |

15:30 | Lorenz Schwachhöfer | Martin Schlichenmaier | Wilfried Schmid | discussions | |

16:30 | Alan Weinstein (zoom) | Niky Kamran | discussions | ||

17:30 | |||||

19:00 | aperitif | dinner IESC |

## Titles and Abstracts

**Mélanie Bertelson**(Université Libre de Bruxelles, BE)

**Title : **Conformal symplectic structures and contact structures

**Abstract :** So-called *locally conformal symplectic *(*lcs*) structures appear naturally when considering contact structures (the odd-dimensional cousin to symplectic structures). In particular, a leafwise lcs form on a codimension one foliation allows to deform the latter into a contact structure. On a closed manifold, lcs forms are quite flexible and exist under minimal conditions, as will be explained in the talk. On codimension one foliations, this is unfortunately not anymore true.

This is a joint work with Gaël Meigniez.

**Pierre Bieliavsky**(Université Catholique de Louvain, BE)

**Title : **Locally compact quantum groups: a geometrical approach to the exact symplectic case (Frobenius Lie groups)

**Abstract : **Symplectic Lie groups are the semi classical limits of formal Drinfeld twists based on universal enveloping algebras. Conversely, constructing a formal Drinfeld twist directed by a symplectic two cocycle on a finite dimensional Lie algebra is an easy task. The non-formal framework is much harder. This is the subject of my talk. Frobenius Lie groups are the “”exact symplectic Lie groups”. Those are the Lie groups that admit open coadjoint orbits. Quantizing a locally compact group consists in finding a dual two cocycle based on the group von Neumann algebra. For symplectic Lie groups, this type of quantization is related to finding some sort of “non-formal star product” on the group. I will report on a geometrical approach to locally compact quantization of Frobenius Lie groups.

This is a joint work with Victor Gayral, Sergey Neshveyev and Lars Tuset.

**Martin Bordemann**(Université de Haute Alsace, FR)

**Title :** Drinfel’d’s associator for the working algebraist refusing to use more analysis than undergraduate level (in collaboration with Andrea Rivezzi and Thomas Weigel, University of Milano-Bicocca, Italy)

**Abstract : **This is a pedagogical talk about the famous associator invented by V.G.Drinfel’d in 1989/90 and a proof of its hexagon and pentagon identities. It is well-known that this associator has applications to the quantization of (quasi) Lie bialgebras (P.I.Etingof-D.A.Kazhdan and B.Enriquez-G.Halbout), deformation quantization (D.E.Tamarkin) and number theory (multiple zeta values). Drinfel’d’s exposition in the original articles (but also in most textbooks) had been quite sketchy, and the mortal reader usually does not to what extent s/he has to know deep complex analysis or algebraic topology. We give a more elementary and explicit approach using limits of (formal) parallel transports along concrete paths in explicit star-shaped domains of the real line and plane with respect to flat (formal) connections derived from the Knizhnik-Zamolodchikov connection.

**Frédéric Bourgeois**(Université Paris-Saclay, FR)

**Title : **A guided tour of Legendrian submanifolds and their homological invariants

**Abstract : **Legendrian submanifolds are of special interest in contact geometry, and their study presents some analogies with knot theory. In particular, various types of invariants are developed in order to distinguish their isotopy classes. This talk will focus on the description of (bi)linearized Legendrian contact homology. Using the structural properties of these invariants, all their possible values can be determined. This will be illustrated with many examples of Legendrian submanifolds.

These results were obtained jointly with Josh Sabloff and Lisa Traynor, and with Damien Galant respectively.

**Henrique Bursztyn**(IMPA, BR)

**Title : **Integration of Poisson homogeneous spaces and shifted symplectic geometry

**Abstract :** A central problem in Poisson geometry concerns the integrability of Poisson manifolds to symplectic groupoids. In this talk, I will show that every Poisson homogeneous space of any Poisson Lie group is integrable in this sense. This result is obtained through a concrete construction and the main tools come from Dirac geometry. I will also explain how this construction can be naturally understood in terms of shifted symplectic geometry. The talk will be based on joint work with D. Iglesias and J.-H. Lu, and work in progress with D. Alvarez and M. Cueca.

**Alberto Cattaneo**(Universität Zürich, CH)

**Title : **Poisson structures from corners of field theories

**Abstract :** The BV formalism and its shifted versions in field theory have a nice compatibility with boundary structures. Namely, one such structure in the bulk induces a shifted (possibly degenerated) version on its boundary. I will discuss in particular how to proceed from the BFV structure on a “space” slice in field theory, which describes the symplectic reduction due to constraints), to a shifted structure on its boundary (the corners of space–time), which in turn describes a Poisson algebra (possibly up to homotopy). I will describe a few examples, including, time permitting, general relativity in the coframe formulation.

**Ezra Getzler**(Northwestern University, US)

**Title : **The Batalin-Vilkovisky formalism for Chern-Simons theories and curved L-infinity algebras

**Abstract :** Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) models are a class of field theories including Chern-Simons theory and the Poisson sigma-model. I explain how, in the Batalin-Vilkovisky formalism, they may be expressed as solutions to a curved Maurer-Cartan equation refining the classical master equation. (There is a variant of our construction in the quantum setting, due to Bonechi-Cattaneo-Qiu-Zabzine.) Using this formalism, we prove that the Lie bracket on the Batalin-Vilkovisky complex for the Chern-Simons theory of a simple Lie algebra is formal (and that the bracket vanishes on the cohomology). As a second application, we explain how Hamiltonian mechanics may be viewed as an AKSZ model: the connection on the prequantization line bundle and the Poisson tensor combine to give a solution of an appropriate curved Maurer-Cartan equation.

**Joel Fine**(Université libre de Bruxelles, BE)

**Title : **Knot invariants from twistor spaces and 6-dimensional symplectic manifolds

**Abstract :** I will describe a programme which aims to define invariants of knots and links in 3-manifolds by studying twistor spaces of 4-manifolds and, more generally, a certain class of symplectic 6-manifolds. Let M be a compact 4-manifold with boundary Y. Suppose we have an asymptotically hyperbolic metric on the interior of M, which is negatively curved. Given a knot or link K in Y, I will explain how one might hope to count minimal surfaces in M with ideal boundary equal to K and, in this way, obtain a link invariant, i.e. the result should only depend on the isotopy class of K. Minimal surfaces in M are in 1-1 correspondence with J-holomorphic curves in the twistor space of M (using the Eells—Salamon almost complex structure). Based on this, one might hope to also count J-holomorphic curves in a more general symplectic 6-manifold X with boundary YxS^2. The J-holomorphic curves should meet the boundary in a curve which projects down to K in Y. For minimal discs in hyperbolic 4-space, this is a theorem and for minimal discs in a general M, I’m close to a proof. I will explain the ideas in the proof and also the difficulties which are still to be overcome in the more general setting.

**Jean Gutt**(Institut National Universitaire Champollion, FR)

**Title : **Symplectic convexity? (an ongoing story…)

**Abstract : **What is the symplectic analogue of being convex? We shall present different ideas to approach this question. Along the way, we shall present recent joint results with J.Dardennes and J.Zhang on monotone toric domains non-symplectomorphic to convex domains and with M.Pereira and V.Ramos on cube-normalized capacities.

**Sonya Hohloch**(Universiteit Antwerpen, BE)

**Title : **Advances in (hyper)semitoric systems

**Abstract : **This talk gives an update on the recent developments around so-called hypersemitoric systems. These are two degree of freedom integrable Hamiltonian systems on 4-dimensional compact symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^{1}-action.

**Niky Kamran**(Université McGill, CA)

**Title : **Uniqueness and stability results for the inverse Steklov problem on Riemannian warped products

**Abstract : **The inverse Steklov problem is a problem of spectral geometry which consists in recovering the metric of a compact Riemannian manifold with boundary from the knowledge of the spectrum of the Dirichlet-to-Neumann map for the Laplacian. We shall consider the inverse Steklov problem on a class of Riemannian warped products which can be thought of as deformations of the closed ball in R^d and shall present several uniqueness and stability results. This is joint work with Thierry Daudé (Cergy-Pontoise) and François Nicoleau (Nantes).

**Laurent La Fuente Gravy**(Université libre de Bruxelles, BE)

**Title : **On formal moment maps

**Abstract : **I will present a construction, using deformation quantization, that recovers and deforms classical moment maps on infinite dimensional spaces. The main tool is the notion of formal connection, introduced by Andersen-Masulli-Schätz, applied to bundles of Fedosov star product algebras.

The examples I will discuss are the Cahen-Gutt moment map on the space of symplectic connections, the Donaldson moment map on diffeomorphisms group and the Donaldson-Fujiki moment map on the space of compatible almost complex structures on a symplectic manifold.

**Camille Laurent-Gengoux**(Université de Lorraine, FR)

**Title :** Two applications of symplectic realizations : integration and desingularization

**Abstract :** It is well known that Poisson manifolds are the base manifolds of symplectic local groupoids. I will present some results of Oscar Cosserat (co-directed by V. Salnikov and myself) using them to construct Poisson integrators, i.e. numerical methods to solve Hamiltonian differential equations, and two related methods to use (sympletic) bisubmersions to make (symplectic) resolutions of singularities for (Poisson) singular foliations, one of which is due to my PhD-student Ruben Louis.

**Eva Miranda**(Universitat Politècnica de Catalunya, ES)

**Title : **From Symplectic to Poisson manifolds and back

**Abstract : **b-Structures and other generalizations (such as E-symplectic structures) are ubiquitous and sometimes hidden, unexpectedly, in a number of problems including the space of pseudo-Riemannian geodesics and regularization transformations of the three-body problem. E-symplectic manifolds include symplectic manifolds with boundary, manifolds with corners, compactified cotangent bundles and regular symplectic foliations. But how general can such structures be? In this talk, I first explain how to associate an E-symplectic structure to a Poisson structure with transverse structure of semisimple type (joint work with Ryszard Nest) and I will connect this to a result by Cahen, Gutt and Rawnsley. This result illustrates how E-symplectic manifolds serve as a trampoline to the investigation of the geometry of Poisson manifolds and the different facets of their quantization. This should let us address a number of open questions in Poisson Geometry and the study of its quantization from a brand-new perspective.

**Sylvie Paycha**(Universtät Postdam, DE)

**Title : **Mathematical reflections on locality

**Abstract :** Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, I will first briefly review some features of the notion of locality arising in physics and mathematics. These are then encoded in locality relations, given by symmetric binary relations whose graph consists of pairs of “mutually independent elements”. Locality morphisms, namely maps that factorise on products of such pairs of elements, play a key role in the context of renormalisation in multiple variables. They include “locality evaluators”, which we use

to consistently evaluate meromorphic germs in several variables at their poles. I will report on recent joint work with Li Guo and Bin Zhang which gives a classification of locality evaluators on certain classes of algebras of meromorphic germs.

**Martin Schlichenmaier**(Université du Luxembourg, LU)

**Title : **Lie superalgebras of Krichever-Novikov type and their central extensions

**Abstract : **Witt and Virasoro type algebras (including their central extensions) are of fundamental importance in Conformal Field Theory. Krichever-Novikov type algebras are their generalisation to higher genus Riemann surfaces. The case of more than two points where poles are allowed is also covered by these more general algebras. We will introduce them in this talk. In a first step we introduce a mother Poisson algebra, which contains the relevant algebras as substructures. Here we concentrate on the superalgebra structure. By quantization we are forced to study central extensions. We classify all almost-graded central extensions. The notion of almost-gradedness will be explained in the talk.

**Wilfried Schmid**(Harvard University, US)

**Title : **Riemann’s paper on Fourier Series

**Abstract : **Bernhard Riemann undertook the first major study of Fourier Series. I shall talk first about his personal life, and then about his results.

**Lorenz Schwachhöfer**(Technische Universität Dortmund, DE)

**Title : **Torsion free connections with prescribed curvature

**Abstract :**This talk is based on joint work with Efrain Basurto. We show that locally, the structure equations of torsion free connections can be linearized, using parallel translation of the curvature tensor along geodesics through a fixed point. As an application, we show that any Berger algebra can occur as the holonomy algebra of a torsion free connection, a result that only was known in special cases so far. We also discuss a question of M. Cahen on torsion free connections which are determined by the covariant derivatives of the curvature up to a fixed order.

**Stefan Waldmann**(Universität Würzburg, DE)

**Title : **Star Products on Cotangent Bundles of a Lie Group

**Abstract : **In this talk I will recall some old construction of Simone Gutt on a canonical star product on T^{*}G for a Lie group G. There are several ways to understand this star product and some variations of it using e.g. symbol calculus or the Fedosov’s construction. I will also indicate how one can obtain convergence results for this star product as recently discussed in a joint work with Michael Heins and Oliver Roth.

**Alan Weinstein**(University of California at Berkeley and Stanford University, US)

**Title : **Hamiltonian Lie algebroids over Poisson manifolds

**Abstract : **Christian Blohmann and I, along with Stefano Ronchi, are continuing a study of hamiltonian Lie algebroids, extending now from symplectic manifolds to Poisson manifolds rather than presymplectic manifolds. Many of the basic ideas remain the same, but there are many new examples, including the cotangent Lie algebroid of a Poisson manifold. (The original work which inspired this, an application to the constraints in general relativity, is still unfinished, but the study of hamiltonian Lie algebroids has taken on a life of its own.) The Poisson case is also the most appropriate setting to test (finally!) some ideas proposed by Alejandro Cabrera, after a talk which I gave in Banff in 2017. These concern the possibility of interpreting the compatibility of a Lie algebroid structure, a connection, and a Poisson structure in terms of a naturally associated pair of Poisson structures on the dual of the Lie algebroid. I will report on work in progress on these problems.

**Marco Zambon**(Katholieke Universiteit Leuven, BE)

**Title : **Multisymplectic observables and higher Courant algebroids

**Abstract : **I will report on joint work with Antonio Miti. Consider a closed, non-degenerate differential form Ω of any degree. Associated to it there is an L-infinity-algebra of observables, and an L-infinity-algebra of sections of the higher Courant algebroid twisted by Ω. Our main result is that there is an L-infinity-embedding of former into the latter, and its components involve the Bernoulli numbers. We display explicit formulae for the embedding, involving the Bernoulli numbers. For symplectic forms this is reduces to a prequantization map, and when Ω is a 3-form the embedding was found by Rogers in 2010.