Schedule

The complete schedule of the SAG Summer school may be viewed in PDF format : SAG Summer School Schedule.

Course description

1.   Cluster categories   (Charles Paquette)

Cluster categories have been defined in 2004 by Buan, Marsh, Reineke, Reiten and Todorov and, independently, by Caldero, Chapoton and Shiffler in the A_n case. They are some triangulated categories defined from acyclic quivers (a quiver is just an oriented graph). Not only cluster categories provide a setting in which one can study a wide class of cluster algebras as defined by Fomin and Zelevinsky in 2001, but they also provide a new background to generalize tilting theory, which is an important subject in representation theory. Since their introduction, cluster categories have been generalized in many different directions, including for quivers with potentials (the quivers need not be acyclic anymore) by Amiot.

This course will give an introduction to cluster categories. We will look at the classical construction for acyclic quivers in details.  After defining cluster-tilting objects and mutations, we will discuss the link between cluster categories and cluster algebras. Time permitting, I will give some insights on how one can generalize cluster categories for arbitrary quivers.

Contents : Triangulated categories, cluster algebras, cluster tilted algebras.

Length : 6 h

Professor : Charles Paquette, University of Connecticut

2.   Combinatorial vector fields  (Tomasz Kaczynski)

Contents :

1. Introduction. Simplicial complex. Homology.
2. Combinatorial vector field and V-paths in the sense of Forman.
3. Extension of V-paths aimed at analysis of asymptotic dynamics.
4. Isolated invariant sets and Morse decomposition.

Length : 3 h

Professor : Tomasz Kaczynski, Université de Sherbrooke

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BIBLIOGRAPHY

T. Kaczynski, M. Mrozek et K. Mischaikow, Computational Homology, Appl. Math. Sci. Vol. 157, Springer, NY 2004; section 11.1.
R. Forman, Combinatorial vector fields and dynamical systems, Mathematische Zeitschrift 1998.
T. Kaczynski, M. Mrozek, and T. Wanner, Towards a formal tie between combinatorial and classical vector field dynamics, J. Comput. Dynamics, 3 (1), (2016), 17-50.

3.   Stability conditions  (Thomas Brüstle)

Stability, as a property of geometric objects, was introduced by Mumford around 1960. It was invented as a tool for the construction of moduli spaces, for example of curves and of vector bundles.  Since their introduction in geometry, such ideas have found traction in many contexts. For example, in representation theory of quivers they occur in semi-invariant theory after their introduction in seminal papers by Schofield  and King in 1990. The semi-invariant picture of quiver representations  has also re-appeared in mathematical physics, for example in Kontsevich and Soibelman's study of wall crossing and Donaldson-Thomas invariants in integrable systems and mirror symmetry in 2014.

This course will give a friendly introduction to various aspects of stability conditions for quivers.

Contents : Representation of quivers, semi-invariants, central charge and stability conditions.

Length : 6 h

Professor : Thomas Brüstle, Université de Sherbrooke and Bishop's University

4.   Algebraic curves   (Vasilisa Shramchenko)

A plane algebraic curve is a zero set of a polynomial in two complex variables. Being an object most frequently studied in algebraic geometry, algebraic curves are omnipresent in modern science with applications ranging from number theory to modern physics and string theory. This course will give a friendly introduction to the theory of algebraic curves with the focus on understanding the topology of the curve starting from its defining equation.

Contents : Singular points of an algebraic curve, Bézout theorem, degree-genus formula.

Length : 6 h

Professor : Vasilisa Shramchenko, Université de Sherbrooke

5.   Random matrix models   (Patrick Labelle)

Quantum field theory is an essential tool in theoretical physics which has also become of central importance in pure mathematics, especially in differential geometry and algebraic topology. These lectures will focus on a few applications of random matrix models, the simplest type of quantum field theories. We will see how they can be used to count certain types of graphs and  how one particular model was key in the proof by Kontsevich of a famous conjecture made by Witten concerning a moduli spaces of curves and their relation to quantum gravity.

Contents : Random matrix models : applications to counting graphs, quantum gravity and topological invariants.

Length : 6 h

Professor : Patrick Labelle, Université de Sherbrooke and Bishop's University

6.   Geometric structures   (Virginie Charette and Son Lam Ho)

A (G,X)-structure on a manifold M means that it looks « locally » like X, with a geometry preserved by G.  Examples include Euclidean structures, projective structures, hyperbolic manifolds, etc.

Our course will give a brief introduction to the theory of (G,X)-structures and will quickly move onto explicit examples.  We will also discuss how the space of all such structures on M can often itself be endowed with a rich geometrical structure.

Contents :

1. Basic example : Euclidean structures on a torus.
2. Theory : developing map and holonomy of a (G,X)-structure.
3. Hyperbolic surfaces and Fricke space.
4. Projective structures on a surface.
5. CP^1 structures.

Length : 9 h

Professors : Virginie Charette and Son Lam Ho, Université de Sherbrooke

Note: The mandatory exercises for every course will be corrected at the end of the Summer School.