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Two base courses.

Seven advanced courses.
Confirmed lectures
Some abstracts and notes (linked to the author) are available (PDF file, updated on Feb. 28)


Other activities:
Talk about "Las matemáticas en el cine", by José Antonio de la Peña.
March 14th, 20hs
Paseo Astor, San Martin y Mitre ( 300 metres from the hotel).










Quelques aspects de base de la théorie de représentations des algèbres.

Ibrahim Assem (Université de Sherbrooke, Québec)

L'organisation du cours sera la suivante:  1. Carquois et algèbres: les algèbres de chemins, les idéaux admissibles et les quotients des algèbres de chemins, le carquois associé à une algèbre de dimension finie;  2.  Représentations et modules: représentations de carquois ayant des relations, les modules simples, les modules projectifs et injectifs;  3. Théorie de Auslander-Reiten: morphismes irréductibles et suites quasi-scindées, la translation de Auslander-Reiten, l'existence de suites quasi-scindées;  4. Le carquois de Auslander-Reiten: la première conjecture de Brauer-Thrall et le théorème de Auslander, un exemple: les algèbres de Nakayama; 5. Théorie basculante: les modules basculants. Le théorème de Brenner-Butler et ses conséquences;  6. Algèbres héréditaires et algèbres basculantes: algèbres héréditaires, carquois de Dynkin et euclidiens, le théorème de Gabriel, les algèbres basculantes et le critère de Liu-Skowronski.






















Polynomial invariants, an introduction to some classical commutative topics and an overview of some current noncommutative developments.

François Dumas (Université de Clermont-Ferrand, France)
The course will be organized in the following way:  1. Linear invariants: actions of subgroups of $GL(n,\mathbb C)$ on polynomial algebras; some classical results in commutative invariant theory; analogues for noncommutative polynomials and Weyl algebras; applications in Lie theory and differential operators algebras. 2.  Multiplicative invariants: actions of subgroups of $GL(n,\mathbb Z)$ on Laurent polynomial algebras; analogues for quantum tori and Hopf algebras; applications in quantum groups theory. 3.  Rational invariants: classical Noether's problem and related results; analogues for noncommutative rational functions; applications to the Weyl skewfields and quantum Weyl skewfields; links with the Gelfand-Kirillov problem. 4. Completion and invariants: automorphisms actions on commutative and non-commutative power series; applications to pseudodifferential operators; links with the modular forms theory.














Generic modules and complexes.

Raymundo Bautista (Universidad Nacional Autónoma de México, Mexico)
The main aim of the course is to present the research made by many authors about properties of  modules of finite endolength, which means that the module has finite length when considered as module over its endomorphism ring.  He will present properties of these modules, and their connection with the pure-injective modules.  The indecomposable not finitely generated finite endolength modules are the so called generic modules.  When the field is algebraically closed and the algebra is of infinite representation type, he will show that there exists a bijection between isomorphism classes of generic modules and families of Auslander-Reiten components, parametrized by one parameter.  Finally, he will present the generic complexes in the derived category of the algebra.














La structure de Poisson de certaines varietés quotient.

Jacques Alev (Université de Reims, France)
Soit V un espace symplectique et G un sous-groupe fini de Sp(V). La variété quotient V/G porte une riche structure de Poisson. Dans ce cours, nous étudierons cette structure et rappellerons les résultats fondamentaux sur les variétés de Poisson. Puis, nous étudierons les déformations non commutatives en vue d'une quantification et de la comparaison des homologies de Poisson de V/G et de l'homologie de Hochschild de sa quantification.














Koszul algebras and applications.

Roberto Martínez-Villa (Universidad Nacional Autónoma de México, Mexico)  
We will present first some basic facts about Koszul algebras and their representations. We will then proceed and talk about the Koszul duality and the Bernstein-Gelfand-Gelfand theorem and some of its uses. One of the main aims of this minicourse is to present examples of situations where Koszul algebras appear or can be used. Among the applications that will be discussed will be the preprojective algebra, selfinjective Koszul algebras and applications to the study of vector bundles over the projective n-space.






















Group actions on finite dimensional algebras and their categories of modules.

José Antonio de la Peña (Universidad Nacional Autónoma de México, Mexico)


We will consider the automorphism groups of algebras and the associated  locally finite categories, and different applications in the representation theory.  We will present the following subjects: 1. Galois coverings: modules of first and second type; invariance of the representation type; 2. Induced actions on the module categories; 3. Symmetries of algebras and their consequences in the theory of associated invariants (quadratic forms, Coxeter polynomial); 4. Computation of the simplicial cohomology and Hochschild cohomology using Galois coverings.



















Derived categories and their applications.

María Julia Redondo (Universidad Nacional del Sur, Argentina) and Andrea Solotar (Universidad de Buenos Aires, Argentina)
We will start with the basic definitions of derived categories, derived functors, tilting complexes and stable equivalences of Morita type. With several examples, we will show that this is the best framework to do homological algebra and will exhibit their usefulness for getting new proofs of well known results. We will consider the invariants of a ring under derived equivalences: Grothendieck group, Hochschild cohomology.























Homological conjectures and degenerations of modules.

Sverre Smalø (University of Trondheim, Norway) 
The first two hours will be dedicated to some homological conjectures and an example giving a negative answer to a question of  Maurice  Auslander. In addition, Prof. Smalø will give a series of lectures on degenerations of modules over the same algebra.

















Cluster categories and their relation to Cluster algebras, Semi-invariants and Homology of torsion free nilpotent groups.

Gordana Todorov (Northeastern University, Boston, USA) 

1. Cluster categories will be defined and their basic properties stated as done by Buan, Marsh, Reineke, Reiten, T.

2. Cluster algebras were introduced by Fomin and Zelevinsky. Known relations between cluster categories and combinatorics of cluster algebras will be stated, as well as some of the open questions.

3. Semi-invariants for quivers were studied by Schofield, Derksen and Weyman. Generalized semi-invariants will be defined and the theorems relating domains of such semi-invariants and the simplicial complexes associated to cluster categories will be given.

4. The same simplicial complexes associated to cluster categories are related to the Igusa-Orr pictures in the homology of nilpotent groups. Results, and many open questions in this direction will be stated.