Abstract: The talk is based on joint work with Sonia Natale. We present the classification of finite dimensional cosemisimple Hopf algebras having a faithful self-dual 2-dimensional comodule (the terminology will be explained during the talk). The classification is done in terms of binary polyedral groups.

Abstract: A finite subgroup G of SL(2,C) is either cyclic (of type A_n), dihedral (type D_n) or one of the three exceptional types (E6,E7,E8). The quotient surface of C^2 by the group G is naturally given in C^3 by an equation F(x,y,z)=0. We will explain how to prove the following result: the extension C(x,y,z)/C(F) is rational (purely transcendental) if and only if G is of type A_n. The proof is related to the action of Galois groups on surfaces.

Abstract: Homological Mirror Symmetry studies the connection between symplectic geometry and algebraic geometry. More precisely it proposes an equivalence between two categories the Fukaya category of a symplectic manifold and the category of singularities of an algebraic singularity. Using noncommutative geometry we can change the last category to the category of matrix factorization of a central element in a certain noncommutative algebra.

In this talk we will show how this relates to certain combinatorial objects called dimer models and show how mirror symmetry translates to a certain duality between these dimer models.

Abstract: In this talk I will explain how to compute the structure ring of the derived self-intersection associated to an inclusion X\subset Y of smooth algebraic varieties, in terms of the universal envelopping algebra of a suitable Lie algebroid. Hopefully, I will have time to give a few applications. This is a work in progress with Andrei Caldararu and Junwu Tu.

Abstract: TBA

Abstract: The Specht modules are important modules defined for the symmetric group in any characrteristic, and a great deal of effort is devoted to finding their structure.  It is known that if the underlying characteristic is not 2, then all Specht modules are indecomposable.  In characteristic 2 there are decomposable Specht modules, but examples are hard to find.  I will report on recent joint work with Craig Dodge, in which we have found new families of decomposable Specht modules.

Abstract: Quantum cluster algebras are a generalisation of the now ubiquitous cluster algebras, the latter being the commutative version of the theory.  The non-commutativity in a quantum cluster algebra is relatively mild: the main feature is that elements in the same cluster must commute up to a power of q, although elements from different clusters may have more complicated relationships.  The study of quantum cluster algebras has recently been boosted by the demonstration by Geiss, Leclerc and Schroer of quantum cluster algebra structures on the quantum coordinate rings of open cells of partial flag varieties, considerably extending the class of examples previously known.  We will discuss the definition of quantum cluster algebras and some of their basic properties, and give an overview of what is known and unknown about these algebras at present.

Abstract: Let G be a finite group, and k a field of nonzero characteristic p. If p divides the order of G, then the group algebra kG is not semi-simple. To each indecomposable k-algebra factor B of kG is associated a G-conjugacy class of p-subgroups of G, called the defect groups of B. Many problems in modular representation theory focus on the relationship between the structure of the defect groups of B and the  structure of B (e.g. the representation type of B is finite if and only if the defect groups of B are cyclic). In this talk, I will present some results which show that the Hochschild cohomology of B is controlled  to a large extent by the defect groups of B. This is joint work with Markus Linckelmann.

Abstract:  The space of polygons in the projective plane modulo the action of PSL(3) can be identified with the space of combinatorial objects called 2-friezes (they are a higher dimension analog of the classical friezes of Coxeter-Conway). I will explain how the two spaces are identified and discuss the main properties of the 2-friezes. In particular I will mention relations with the cluster algebras. This is a joint work with V.Ovsienko and S.Tabachnikov.

Abstract: Recently, Assem, Reutenauer and Smith have introduced families of sequences associated to the vertices of an acyclic quiver Q. These sequences consist of cluster variables. They proved that if the sequences associated with Q satisfy linear recurrence relations, then Q is necessarily affine or Dynkin. Conversely, they conjectured that, the sequences associated with a quiver of Dynkin or affine type always satisfy linear recurrence relations.  In my talk I will present a proof of the Assem-Reutenauer-Smith's conjecture using the representation-theoretic approach to cluster algebras. More precisely, our main tool is the categorification of acyclic cluster algebras via cluster categories. This is joint work with Bernhard Keller.