Speakers

During each of the two weeks of the school, there will be three courses of 5 hours. They will be given by the following leading experts:

Sergey Fomin (University of Michigan)

Mark Gross (UC San Diego)

Maxim Kontsevich (IHES)

Grigory Mikhalkin (University of Geneva)

Michael Shapiro (Michigan State University)

David Speyer (Univeristy of Michigan)

Timetable

The timetable could still be subject to possible modifications.

Second Week

It is also downloadable here.

Notice that Wednesday afternoons will be free. On the afternoon of Wednesday, April 18, a presentation of the Sage cluster algebra package by Christian Stump (Hannover) will take place at Chevaleret.

Programme of the courses

Sergey Fomin

Cluster algebras: basic notions and key examples.(Slides: Lectures 1-3)

Lectures 1-2: Fundamentals of cluster algebras. First examples.
Lecture 3: Cluster algebras and triangulated surfaces.
Lecture 4-5: Cluster structures on some classical rings of invariants.

Mark Gross

Mirror symmetry for surfaces and tropical geometry.

I will discuss a circle of ideas given in joint work with Paul Hacking and Sean Keel. I will explain how tropical geometry and Gromov-Witten theory can be used to construct mirrors to rational surfaces along with a choice of an anti-canonical cycle of rational curves. I will explain in detail some of the key ingredients of this construction, including the tropical vertex, as developed by myself with Pandharipande and Siebert, and the construction of theta functions, again using tropical techniques, in the context of mirror symmetry. If there is time, I hope to explain connections between these constructions and cluster varieties.

Maxim Kontsevich

Integrable systems, wall-crossings and asymptotics

The goal of my lectures is to explain mathematics behind the works of physiscits D.Gaiotto, G.Moore and A. Neitzke, arXiv: 0807.4623 and 0907.3947. With a generic point of the base of Hitchin integrable system one can associate canonically a formal coordinate system near a cusp of the moduli space of local systems. This coordinate system is adapted to the study of the asymptotic behavior of the monodromy for equations depending on small parameter. Crossing the walls on the base gives rise to a coordinate change, which is the usual cluster transformation in certain regions. Several aspects of this picture generalize to mutations of arbitrary quivers.

Grigory Mikhalkin

Lecture 1: Points, functions, forms and varieties under the tropical limit.
Lecture 2: Tropical manifolds: definitions and examples.
Lecture 3: Curves, linear systems, tropical Brill-Noether theory.
Lecture 4: Tropical enumerative geometry.
Lecture 5: Further topics.

Michael Shapiro

Lecture 1: Introduction to Poisson formalism (slides)
Lecture 2: Cluster algebras and compatible Poisson structure (slides)
Lecture 3: Examples and applications (e.g., Weil-Petersson form and “cluster determines seed”-statement)(slides)
Lecture 4: Planar networks(slides)
Lecture 5: Pentagram maps from cluster perspective(slides)

References:

1. Arnold “Mathematical Methods of Classical Mechanics”, New York : Springer-Verlag, c1989
2. Gekhtman, Shapiro, Vainshtein “Cluster algebras and Poisson Geometry”, Providence, R.I. : American Mathematical Society, 2010.
3. Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein arXiv:1110.0472 Higher pentagram maps, weighted directed networks, and cluster dynamics
4. M. Gekhtman, M. Shapiro, A. Vainshtein, arXiv:math/0208033 Cluster algebras and Poisson geometry
5. M. Gekhtman, M. Shapiro, A. Vainshtein, arXiv:math/0309138 Cluster algebras and Weil-Petersson forms
6. M. Gekhtman, M. Shapiro, A. Vainshtein, arXiv:math/0703151 On the properties of the exchange graph of a cluster algebra
7. M. Gekhtman, M. Shapiro, A. Vainshtein, arXiv:0805.3541 Poisson Geometry of Directed Networks in a Disk
8. M. Gekhtman, M. Shapiro, A. Vainshtein, arXiv:0901.0020 Poisson Geometry of Directed Networks in an Annulus

David Speyer (Slides)

Lecture 1: Introduction to tropical geometry
Lecture 2: Positive tropical geometry
Lecture 3: Tropicalizations of cluster algebras -- examples
Lecture 4: Tropicalizations of cluster algebras -- the conjectures of Fock and Goncharov
Lecture 5: Tropical geometry and compactifications

Abstracts of talks

Paris, April 16-27, 2012

Jussieu Campus , Esclangon Building, Herpin Lecture Hall