This advanced school is part of the

Joint CRM-ICMAT Intensive Research Programme Poisson 2022

The Summer School on Poisson Geometry aims to introduce the participants to Poisson Geometry and related fields, as well as to present some of the most recent developments and open problems in the field. The targeted audience is final-year undergraduate students, graduate students and young researchers. The only required background is basic differential geometry.

The school will consist of four minicourses during one week. There will be problems sessions to promote the participation of the students of the school, and also several activities to foster social interaction.

This school will be followed by a week-long conference which will take place in Madrid.

Poisson Geometry lies on the crossroads of Mathematical Physics and Geometry. It originates in the mathematical formulation of Hamiltonian mechanics and as the semiclassical limit of quantum systems. Although Poisson structures can be traced back to some of the classical works of the 19th century (Poisson, Hamilton, Jacobi, Lie), Poisson Geometry started as a field around 1980 with the work of Lichnerowicz, Weinstein, etc. The field developed rapidly, stimulated by the connections with a large number of areas in mathematics and mathematical physics, including differential geometry and Lie theory, quantization, noncommutative geometry, representation theory and quantum groups, geometric mechanics, and integrable systems.

Wonderful Compactifications

Lecturer: Ana Balibanu (Harvard)

Person in charge of the problems sessions: Yu Li (MPIM, Bonn), Problem List WC

Chair: Marco Gualtieri (Toronto)

Abstract: Every complex semisimple group of adjoint type has a canonical, smooth compactification which encodes the asymptotic behavior of the group “at infinity.” It is constructed using representation-theoretic data, and its geometry is indexed by the combinatorics of the corresponding root system. This course will outline some of the remarkable properties of this algebraic variety, and some of its many applications.

Lecture 1: Outline of the construction. Structure theory of the wonderful compactification, its boundary divisor, and distinguished subvarieties.

Lecture 2: Poisson structures associated to the compactification. Poisson homogeneous spaces and the variety of Lagrangian subalgebras.

Lecture 3: Current developments, additive analogues, and connections to combinatorics and algebraic geometry.

Introduction to Poisson Geometry

Lecturer: Henrique Bursztyn (IMPA), Slides IPG

Person in charge of the problems sessions: Pedro Carvalho (IMPA), Problem List IPG

Chair: Eckhard Meinrenken (Toronto)

Abstract: This mini-course will offer a panoramic view of Poisson geometry, including both classical aspects and modern developments of the theory.

We will start with basic definitions and concepts related to Poisson structures, some foundational results (local theory, symplectic leaves) and key examples. We will then discuss symplectic groupoids and more general connections with Lie groupoids and multiplicative geometric structures. I will also introduce Dirac structures, their main properties and various applications, both within and beyond Poisson geometry.

Singular Foliations

Lecturer: Camille Laurent-Gengoux (Metz)

Person in charge of the problems sessions: Ruben Louis (Univ Lorraine)

• Problem List SF
• Slides Problem Sessions
• Solutions Problem Sessions

Chair: Rui Fernandes (Urbana-Champaign)

Abstract: Foliations are partitions of a manifold into sub-manifolds called leaves. They are said to be singular when the dimension all some leaves are not constant. Orbits of Lie group actions, symplectic leaves of a Poisson bivector, symmetries of affine varieties are among examples. I will try to give a detailed proof of several basic but subtle theorems about singular foliations and to give an overview of the geometry of their singular leaves.

Lecture 1: Discussion of the various possible definitions of singular foliation. Examples and counter examples. Existence of leaves and local splitting theorem.

Slides Lecture 1

Lecture 2: Various algebraic or geometric structures hidden behind a singular foliation: holonomy Lie algebras, holonomy groupoids. Bi-submersions and first-return.

Slides Lecture 2

Lecture 3: Open (and closed) questions from algebraic and differential geometry to theoretical physics.

Slides Lecture 3

A problem session will take place after Lecture 2.

Geometric quantization, old and new

Lecturer: Jonathan Weitsman (Northeastern University)

Person in charge of the problems sessions: Pau Mir (UPC), Problem List GQ

Chair: Chenchang Zhu (Goettingen)

Abstract: We give an overview of Geometric Quantization, with an emphasis on examples and on invariance of polarization, and on applications to analysis and representation theory.  We then study recent extensions of the theory which apply to the a class of Poisson manifolds, the b-symplectic manifolds, when equipped with Hamiltonian torus actions.

You can register for both activities, the Poisson Advanced School taking place at the CRM from 18-22 July 2022 and the Poisson Conference, taking place at the ICMAT from 25-29 July 2022.

COMBINED REGISTRATION FEE: 300€

The combined registration covers:

• • For the Advanced Course: reception, coffee breaks, daily lunch and a guided visit to Barcelona.
  • For the Poisson 2022 Conference: reception, coffee breaks and conference dinner.

* If you wish to apply for a grant for the Advanced Course, you will need to register separately through this website and through the conference website (click here)

• • Application deadline is May 15, 2022
  • Resolutions will be sent by June 02, 2022

* grants cover everything included with the registration (reception, coffee breaks, daily lunch and a guided visit to Barcelona).

Mario García Fernández (ICMAT–UAM)

David Iglesias Ponte (La Laguna)

David Martín de Diego (ICMAT–CSIC)

Anastasiia Matveeva (UPC)

Eva Miranda (UPC-CRM) – Chair

Cédric Oms (UPC)

Daniel Peralta-Salas (ICMAT–CSIC)

Roberto Rubio (UAB)

Chenchang Zhu (Goettingen) – Chair

Anton Alekseev (Geneva)

Henrique Bursztyn (IMPA)

Rui Loja Fernandes (Urbana-Champaign)

Eckhard Meinrenken (Toronto)

Jiang-Hua Lu (Hong Kong)

IF YOUR INSTITUTION COVERS YOUR REGISTRATION FEE: Please note that, in case your institution is paying for the registration via bank transfer, you will have to indicate your institution details and choose “Transfer” as the payment method at the end of the process.

If you are not applying for a grant, here you can find some options regarding accommodation during the event.