Courses

The Summer School will consist of 4 minicourses. Click here to see the school schedule.

**Location:** Ruppert Blauw lecture room in the Educatorium building in the Utrecht Science Park (de Uithof). See map.

Poisson Geometry- Global Aspects

**Lecturer:** Rui Loja Fernandes

**Length:** 4 x 1 hour

We will start with a quick introduction to some basis concepts in Poisson geometry, such as complete Poisson maps, symplectic realizations, coisotropic submanifolds, etc.. Then we move to some recent topics such as coisotropic deformations, stability of leaves, normal forms around leaves, rigidity.
**Lecture Notes: PDF 1** **PDF 2** **PDF 3**
**Videos: Lecture 1** **Lecture 2** **Lecture 3** **Lecture 4**

Poisson and Symplectic Geometry of Moduli Spaces of Flat Connections

**Lecturer:** Anton Alekseev

**Length:** 4 x 1 hour

Moduli spaces of flat connections on surfaces are among the most interesting examples of Poisson and sympelctic spaces. We'll review finite
and infinite-dimensional constructions of symplectic forms and Poisson bivectors, and give a brief introduction into the following topics:

1) Symplectic volumes and Duistermaat-Heckman localization;

2) Moduli of flat connections and integrable systems;

3) Quantization of moduli spaces and Verlinde formula.
**Lecture Notes: PDF 1** **PDF 2** **PDF 3**
**Videos: Lecture 1** **Lecture 2** **Lecture 3** **Lecture 4**

Lie Groupoids and Multiplicative Structures

**Lecturer:** Henrique Bursztyn

**Length:** 4 x 1 hour

The global objects underlying Poisson structures, encoding relevant symmetries, are symplectic groupoids, i.e., Lie groupoids equipped with multiplicative symplectic structures; they play a key role, for instance, in the Poisson-geometric approach to the theory of moment maps. In this course we will give a short introduction to Lie groupoids and symplectic groupoids, followed by a more general discussion of multiplicative structures that are relevant in Poisson geometry.

**Outline of Course: PDF**
**Videos: Lecture 1** **Lecture 2** **Lecture 3** **Lecture 4**

Cluster Algebras and Compatible Poisson Structures

**Lecturer:** Michael Gekhtman

**Length:** 4 x 1 hour

After reviewing basic definitions in the theory of cluster algebras and discussing several examples, I will concentrate on the notion of the Poisson brackets compatible with the cluster structure and explain how it can be used to recognize cluster structures in coordinate rings of Poisson varieties. Examples will include Grassmannians, Teichmueller spaces and Poisson-Lie groups. A connection with the theory of integrable systems and Poisson geometry of networks on surfaces will also be discussed.
**Lecture Notes: PDF**
**Videos: Lecture 1** **Lecture 2** **Lecture 3** **Lecture 4**