[ Français ]

Cluster algebras, Poisson structure, Networks and Integrable systems

Two series of minicourse lectures given by:
Prof. Michael Gekhtman (University of Notre Dame)

I: July 6-10, 2015 : Cluster structures on Poisson manifolds

2:00 pm, room 4336
Pavillon André-Aisenstadt
Université de Montréal

Lecture 1.

Monday, July 6, 2015
Definition and examples of cluster algebras and compatible Poisson brackets

Lecture 2.

Tuesday, July 7, 2015
Cluster structures on double Bruhat cells and Grassmannians

Lecture 3.

Thursday, July 9, 2015
Cluster algebras from surfaces

Lecture 4.

Friday, July 10, 2015
Cluster structures on Poisson-Lie groups

II: July 20-24: : Networks on surfaces and integrable systems

11:00 am, room 5340
Pavillon André-Aisenstadt
Université de Montréal

Lecture 1.

Monday, July 20, 2015
Network parametrization of totally nonnegative cells in Grassmannians

Lecture 2.

Tuesday, July 21, 2015
Poisson geometry of directed networks in a disc and a cylinder

Lecture 3.

Thursday, July 23, 2015
Q-systems, Coxeter-Toda lattices and cluster algebras

Lecture 4.

Friday, July 24, 2015
Networks on a torus and complete integrability of generalized pentagram maps

The lectures will be of 1 hour duration. The detailed schedule will be posted on the programme’s website.

Useful references:

1. A. Postnikov, Total positivity, Grassmannians, and networks,

2. M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, AMS Mathematical Surveys and Monographs, volume 167 (2010), chapters 2, 4, 8.

3. Y. Kodama, L. Williams, KP solitons and total positivity for the Grassmannian, Invent. Math. 198 (2014), pp. 637-699.

4. R. J. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, EMS 2013.

5. A. Goncharov, R. Kenyon, Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 747–813.

6. M. Gekhtman, M. Shapiro, S. Tabachnikov, A. Vainshtein, Integrable cluster dynamics of directed networks and pentagram maps, arXiv:1406.1883.

This minicourse is part of the program culminating in the workshop: Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes", to take place at the CRM July 27-31, 2015.

Participation is open, but registration is required in order to receive materials related to the series, make use of CRM facilities.


Further information:

J. Harnad, Director,
Mathematical Physics Laboratory
Centre de recherches mathématiques
Université de Montréal
C.P. 6128, Succ. Centre ville
Montréal, Québec, H3C 3J7