Abstract: In this talk we shall consider twisted category algebras over fields of characteristic 0. The underlying category will always be finite and will have an additional property, which is called `split'. The multiplication in such an algebra is essentially induced by the composition of morphisms in the category. Prominent examples of twisted category algebras are various classes of diagram algebras (for suitable parameters) such as Brauer algebras, Temperley--Lieb algebras, or partition algebras. Twisted category algebras also arise in connection with double Burnside rings and biset functors.

We shall show that a twisted split category algebra in characteristic 0 is quasi-hereditary, that is, the corresponding module category is a highest weight category. Moreover, we shall give an explicit description of its standard modules with respect to a particular partial order on the set of isomorphism classes of simple modules. This provides, in particular, a unified proof of the known fact that the aforementioned diagram algebras are quasi-hereditary.

This is joint work with Robert Boltje.

Abstract: The classical Rankin-Cohen brackets are a sequence RC_n of bifferential operators defining a star product on the algebra M_{\star} of modular forms. In a joint work with Emmanuel Royer, we construct star products on the algebra M_{\star}^{\infty} of quasimodular forms, using a classification of the Poisson structures on M_{\star}^{\infty} whose restriction to M_{\star} is RC_1.

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Abstract: Mutation of torsion pairs in triangulated categories and its combinatorial interpretation for cluster categories of Dynkin type A have been studied by Zhou and Zhu. In this talk we present a combinatorial model for mutation of torsion pairs in cluster categories of Dynkin type D, using Ptolemy diagrams of type D, which were introduced by Holm, Jorgensen and Rubey.

Abstract: A Nichols algebra is a universal quotient of the free non-commutative algebra generated by a braided vector space. It has the structure of a braided Hopf algebra. Important features of Nichols algebras can be explained using a combinatorics which is familiar in a (super) Lie theoretic context. In the talk, the origin of this combinatorics is explained, and the main results and recent progress in the theory are sketched.

Abstract: It is well-known that the representation theory of the symmetric group is related to the combinatorics of partitions in both the ordinar and the modular case. The aim of this talk is to show that, in the context of the Weyl group of type B and its Hecke algebra, the notion of Lusztig symbols play a similar role. We present several results around this notion and we show how they can be possibly generalized to a class of complex reflection groups.

Abstract: SL_2-tilings were introduced by Assem, Reutenauer, and Smith in connection with friezes and their applications to cluster algebras. An SL_2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2x2-submatrix has determinant 1. We construct a large class of new SL_2-tilings which contains the previously known ones. More precisely, we show that there is a bijection between our class of SL_2-tilings and certain combinatorial objects, namely triangulations of the strip.

Abstract: The preprojective algebras are non-commutative algebras derived equivalent to crepant resolutions of Kleinian singularities. They have graded deformations, the deformed preprojective algebras, constructed by Crawley-Boevey and Holland. These encode deformations in the geometry, are the simplest case of symplectic reflection algebras, and are of interest in many areas.

The reconstruction algebras are derived equivalent to minimal resolutions of arbitrary quotient surface singularities,and hence a natural generalisation of preprojective algebras. I will consider their graded deformations, generalising the deformed preprojective algebras outside of the Kleinian case.

Abstract:  Jacobian algebras are defined via quivers with potentials and have been used by Derksen, Weyman and Zelevinsky to categorify cluster algebras. They also appear in various other contexts. We show that the representation type of a Jacobian algebra is closely related to the mutation type of its quiver.

This is joint work with Christof Geiss and Daniel Labardini-Fragoso.

Abstract: In this talk, I will describe some invariants of stable equivalence of Morita type related to Hochschild cohomology (results due to Le, Pogorzaly, Xi, Zhou, Zimmermann in particular). I shall then use these invariants to separate some algebras up to stable equivalence of Morita type. Part of these results are joint with Nicole Snashall.

Abstract:  The finite W-algebra is a filtered deformation of the coordinate algebra of the Slodowy slice associated to a nilpotent orbit in the Lie algebra of a reductive group. Their representation theory has important applications to modular representation theory of Lie algebras, some of which I will describe. I will then discuss my joint work with Premet, in which the geometry of sheets is shown to control the classification of the one dimensional representations of the finite W-algebra associated to a classical group.

Abstract: An affine toric variety is an affine algebraic variety with an open dense set T isomorphic to an algebraic torus, with the action of T over itself extending to the whole variety. Since much is known about toric varieties, there has been a lot of interest in the deformation of varieties to toric varieties. In this talk we will discuss a quantum analogue of toric varieties, which turn out to have "geometrical" regularity properties similar to those of their classical counterparts. Just as in the commutative case, one can find quantum "toric degenerations" for some noncommutative algebras; we will finish with some applications of this idea to the study of quantum (deformations of the coordinate rings of) grassmanians in type A and related quantum varieties.