• Ernst Dieterich (Uppsala University), Two-Dimensional Real Division Algebras

       Abstract: A famous theorem of Hopf, Kervaire, Bott and Milnor asserts that every real division algebra has dimension 1,2,4 or 8. Trying to classify (up to isomorphism) all real division algebras of fixed dimension d, it turns out that this problem has a trivial solution in case d = 1, while it is still wide open in the cases d = 4 and d = 8. Even in the seemingly innocent case d = 2 only a few stray attempts to solve it are published, with varying degree of insight. I will present a recently proved structure theorem asserting that the category of all 2-dimensional real division algebras splits into four full subcategories each of which is given by the natural action of a group of order 2 or 6 (dihedral) on the set of all pairs of ellipses in the Euclidean plane which are centered in the origin and have reciprocal axis lengths. From this structural insight a classification of all 2-dimensional real division algebras can be derived fairly easily.

• Pavel Etingof (MIT), Hochschild cohomology of preprojective algebras

       Abstract: Preprojective algebras of quivers were introduced by Gelfand and Ponomarev as a technical tool of studying quiver representations. More recently it became clear that these algebras are very interesting objects by themselves. Namely, their varieties of representations, called quiver varieties, turned out to play a central role in geometric representation theory,
and consequently were studied in detail by many people, notably W. Crawley-Boevey, G. Lusztig, and H. Nakajima.

I will talk about Hochschild cohomology of preprojective algebras A, which was computed in a work by Crawley-Boevey, Ginzburg, and myself (excluding the finite and affine Dynkin cases).
First of all, it turns out that the cohomology is concentrated in degrees 0,1,2, (in particular, the deformations of A are unobstructed). Next, we show that H^0(A)=C, H^1(A)=CE*(L/R), and H^2(A)=L, where L=A/[A,A], R is the degree zero part of A and L(spanned by the vertex idempotents), and E is the derivation of L induced by the grading of A. The space L has a Lie algebra structure ("necklace bracket"), which determines the bracket on H^1(A) and more generally the structure of a Gerstenhaber algebra on the total cohomology H^*(A). Finally, we calculate the Hilbert series of L. The proofs are based on the theory of quiver varieties and the theory of random unitary matrices of large size.

• Martin Herschend (Uppsala University), Solution to the Clebsch-Gordan problem for representations of a class of extended Dynkin quivers

       Abstract: The Clebsch-Gordan problem originates from the setting of representations of the special unitary group. There the solution both is used to classify all indecomposable representations and gives an explicit formula, called the Clebsch-Gordan formula, for the decomposition of the tensor product of any two representations. We consider the Clebsch-Gordan problem for quiver representations, where a tensor product is defined point-wise and arrow-wise. Over an algebraically closed ground field of characteristic 0, this problem has been solved for the loop by B. Huppert and independently by A. Martsinkovsky and A. Vlassov. It has also been solved for some representations of the four factorspace quiver by E. Dieterich in connection with his investigation of lattices over curve singularities. I will present the solution and indicate the proof for a larger class of extended Dynkin quivers.

• Mark Kleiner (Syracuse University), The graded preprojective algebra of a quiver and almost split sequences with preprojective terms

       Abstract: We introduce a grading on the preprojective algebra of a locally finite quiver and, using the grading, construct a collection of canonical exact sequences inside the algebra.  When the quiver is finite and without oriented cycles, the canonical sequences are almost split sequences with preprojective terms.  We obtain a geometric description of the indecomposable preprojective modules and irreducible maps between them; in particular, each such irreducible map is the multiplication by a scalar multiple of an appropriate arrow of the extended quiver.

• Helmut Lenzing (University of Paderborn), Tubular and elliptic curves

       Abstract: Let k be an algebraically closed field. Let T be a tubular curve, that is, a weighted projective line of weight type (2,2,2,2), (3,3,3), (4,2,2) or (6,3,2). It has been known for a long time that the category of coherent sheaves over T is very similar to the category of coherent sheaves over an elliptic curve.

Theorem. The category of coherent sheaves on a tubular or an elliptic curve is a hereditary, noetherian, Hom-finite k-category with Serre duality such that the Auslander-Reiten translation  has finite period. Conversely, each such category is equivalent to the category of coherent sheaves over a tubular or an elliptic curve, where the period of the Auslander-Reiten translation determines which of the two cases happens.

Moreover, we establish a natural map from (isomorphism classes) of tubular curves to (isomorphism classes) of elliptic curves. This map is surjective and generically bijective, and we relate the categories of coherent sheaves for corresponding curves.

• Shiping Liu (University of Sherbrooke), Some homological conjectures for quasi-stratified algebras

       Abstract: This is a joint work with Charles Paquette. We introduce a new class of algebras, called quasi-stratified algebras, that includes the standardly stratified algebras. We shall establish the Cartan determinant conjecture and the strong no loop conjecture for this class of algebras.

• Izuru Mori (SUNY, Brockport), A triangulated  category satisfying Serre duality.

       Abstract: We will formulate a Riemann-Roch like theorem for triangulated categories satisfying Serre duality.  As an application, we will prove that the Riemann-Roch and Adjunction Formula can be extended to noncommutative Cohen-Macaulay surfaces in terms of sheaf cohomology.  We will also prove that these formulas hold for the stable categories over AS-Gorenstein Koszul connected graded algebras in terms of Tate-Vogel cohomology, by extending the BGG correspondence.

• Ralf Schiffler (Carleton University), Quivers with relations arising from clusters.

       Abstract: This is joint work with P.Caldero and F.Chapoton. Given a finite cluster algebra of type A, one can associate a quiver with relations to each cluster such that the indecomposable representations of the quiver are in bijection with the cluster variables that do not lie in that cluster. In this talk, I will define the cluster quiver and its relations in type A and explain how to write down its Auslander-Reiten quiver. If time permits, I will expose some recent progress on a conjecture of Buan, Marsh, Reineke, Reiten and Todorov, relating their tilting objects to our cluster quivers.

• Gordana Todorov (Northeastern University), TBA
  

• Rita Zuazua (Institute of Mathematics, UNAM, Morelia), Auslander-Reiten formula in C_n(P).

       Abstract: Let P be the category of projective A-modules for A an Artin algebra. In this talk we show an Auslander-Reiten formula in the category of projective finite complexes C_n(P).