Corrado De Concini (Roma),

Covariants in the exterior algebra of a simple Lie algebra

Michel Dubois-Violette (Paris),

The quantum group of a preregular multilinear form

Eric Hoffbeck (Paris),

Proving Koszulness using linear polygraphs

Some homological properties of monoids can be proved using "polygraphs" (higher categories, built recursively, whose goal is to encode the geometry of relations of the monoid, of relations between relations, and so on).
In our work, we define a generalization of this notion: linear polygraphs. Roughly speaking, they are higher categories, enriched over vector spaces. Starting from an algebra A given by generators and relations, we define the notion of a confluent presentation of this algebra and we construct a linear polygraph associated to such a presentation. This linear polygraph induces a resolution of the base field by A-modules, with more flexibility than the resolutions obtained via Gröbner bases and Anick chains.
We will show on several examples how these linear polygraphs work, and how they can help us prove that an algebra is Koszul (or N-Koszul).
This is a joint work with Yves Guiraud and Philippe Malbos.

Anthony Joseph (Rehovot),

The integrality of adapted pairs

The notion of an adapted pair is rather useful in the construction of a normal form for regular orbits.  The co-adjoint case is particularly interesting, especially when there are no proper semi-invariants and the invariant ring is polynomial.   Conjecturally the semisimple element of an adapted pair should in this case only admit integer eigenvalues on the Lie algebra.  Some techniques for proving this are developed and applied to truncated biparabolic subalgebras of a semisimple Lie algebra. 

Christian Kassel (Strasbourg),

Le problème de Noether étendu aux algèbres de Hopf

Soit G un groupe fini et k un corps commutatif. Considérons l'extension transcendante pure K de k engendrée par des indéterminées t(g) indexées par les éléments du groupe. Le groupe G opère sur K par h • t(g) = t(hg). Dans un article publié en 1917 Emmy Noether s'est posé la question de savoir si le sous-corps des éléments G-invariants de K est lui aussi une extension transcendante pure de k. On sait maintenant que la réponse peut être positive ou négative ; elle dépend à la fois du groupe G et du corps de base k. Dans un travail récent avec Akira Masuoka (Tsukuba), j'ai étendu le problème de Noether aux algèbres de Hopf de dimension finie. Nous montrons que pour une certaine classe d'algèbres de Hopf la réponse au problème de Noether généralisé est toujours positive.

Thierry Lambre (Clermont-Ferrand),

Dualité des calculs de Tamarkin-Tsygan pour divers types d’algèbres

Les structures de Batalin-Vilkovisky  (BV) ont été introduites  en 1985 par J.-L. Koszul en homologie de Poisson, puis en topologie des cordes  par Chas et Sullivan en 1999, et encore plus récemment en homologie de Hochschild, par V. Ginzburg en 2006,  qui dans son article fondateur sur les algèbres de Calabi-Yau montre que les algèbres Calabi-Yau sont BV. Depuis,  ces  structures BV ont été intensivement étudiées par divers auteurs : T. Tradler, J. Huebschmann, Kowalzig et Krämer, Zhou et Zimmermann, etc. Nous montrerons comment la notion de calcul de Tamarkin-Tsygan  permet de construire algébriquement des structures BV pour des algèbres  à dualité, au sens de Van den Bergh.

Anne Pichereau (Saint-Etienne),

A Calabi-Yau algebra, deformation of a Poisson algebra

This is a joint work with Roland Berger. Following the idea that classical mechanics should be a limit case of quantum mechanics, P.A.M. Dirac explained that the commutator of dynamical variables in quantum mechanics should be the analogue of the symplectic Poisson bracket of |R^{2r} in classical mechanics.                                                                             
Working in a mathematical setting, we consider a non-commutative algebra B, which can be seen as a deformation of a Poisson algebra S.
This algebra B belongs to a family of 3-Calabi-Yau algebras defined by potentials and depending on a natural integer n and the algebra B was for us the most interesting example in the case n=2. We give cohomological links between B and S, as we obtain the Poisson cohomology of S and prove that the Hochschild cohomology of B is isomorphic to the Poisson cohomology of S. In this talk, I will introduce the algebras B and S, explain this last result and show what could be done beyond...

Markus Reineke (Wuppertal),

Arithmetic of character varieties of free groups

Via counting over finite fields, we derive explicit formulas for the E-polynomials and Euler characteristics of GL(d)- and PGL(d)-character varieties of free groups. We prove a positivity property for these polynomials and relate them to the number of subgroups of finite index.

Andrea Solotar (Buenos Aires),

A criterion for homogeneous potentials to be 3-Calabi-Yau applied to algebras constructed from Steiner triple systems

Rachel Taillefer (Clermont-Ferrand),

On generalisations of N-Koszul algebras for Brauer graph algebras

Koszul algebras are a well-known and much studied class of algebras. These were generalised in 2001 by Roland Berger to N-Koszul algebras. This means that if we write the algebra as a quotient of a tensor algebra A=T_k(V)/I, the ideal I can be generated by elements of degree N and that the projective modules in a minimal graded projective resolution of k can be generated in specific degrees depending on N.
Moreover, the Ext algebra of k is generated in degrees 0, 1 and 2.

This notion has been generalised since in several ways. We are interested in two of them:
  - an algebra is called K_2 if it is graded and if its Ext algebra is generated in degrees 0, 1 and 2 [Cassidy-Shelton];
  - an algebra A=T_k(V)/I is called 2-d-determined if the ideal I can be generated by elements of degrees 2 and d, where d>2 is an integer,  and the projective modules in a minimal graded projective resolution of k can be generated in specific degrees depending on 2 and d [Green-Marcos].

The aim of this talk is to give examples of such algebras, within the class of Brauer graph algebras, and to compare K_2 Brauer graph algebras and 2-d-determined Brauer graph algebras.

This is joint work with E.L. Green, S. Schroll and N. Snashall.

James Zhang (Seattle),

Zariski cancellation problem for noncommutative algebras.

Discriminant is used to solve Zariski cancellation problem for a family of noncommutative algebras. Joint work
with Jason Bell.