Schedule
October 6.
Location:
320 Behrakis
1:30 - 2:10 J. Weyman, Semi-invariants
of quivers: new
developments.
2:10 - 2:30 Coffee
break
2:30 - 3:10 M. Neusel, Rings
of invariants satisfying the Weak
Splitting Principle.
3:30
- 4:10 D. Happel, TBA
October 7.
Location:
315 Behrakis
9:30 - 10:10
I. Horozov, Multiple
zeta functions, modular forms and
iteration over the adeles.
10:30 - 11:10 R. Schiffler, Geometric realizations of
cluster categories.
11:30 - 12:10 T.
Bruestle, Cyclic cluster algebras of
rank three.
2:00 - 2:40 G. Todorov, m-cluster categories
3:00 - 3:40
M. Kleiner, Representations
of quivers and the Weyl group of a Kac-Moody algebra.
Titles and Abstracts
Thomas
Bruestle, Cyclic cluster algebras of rank three
Abstract: Cluster algebras, introduced by Fomin and Zelevinsky a few
years ago, have gained a lot of interest by now. Acylic cluster
algebras have been shown to be
related to cluster categories and tilted algebras. Cyclic cluster
algebras, however, are less well understood. We consider the first
non-trivial case, cluster algebras of rank three (square,
coefficient-free), and study which of them are cyclic. Rank three
cluster algebras are given by triples of integers (x,y,z), and we
provide an answer which involves the hyperplanes defined by x2 + y2 +
z2 - xyz = c.
Ivan Horozov, Multiple zeta functions, modular forms and
iteration over the adeles
Abstract: Multiple zeta functions with values at the positive integers
have an interpretation as iterated integrals, due to Kontsevich. We
give an intepretation of multiple zeta functions with complex variables
whose real part is greater than 1 as iterated integrals over the
adeles. Iteration of modular forms was recently defined by Manin. We
give interpretation of iterated modular forms as iteration over the
adeles.
Mark
Kleiner, Representations
of quivers and the Weyl group of a Kac-Moody algebra
Abstract: We discuss connections between the preprojective
representations of a quiver, the (+)-admissible sequences of vertices,
and the Weyl group. To each preprojective representation
corresponds a canonical (+)-admissible sequence. A (+)-admissible
sequence is the canonical sequence of some preprojective representation
if and only if the product of simple reflections associated to the
vertices of the sequence is a reduced word in the Weyl group. As
a consequence, for any Coxeter element of the Weyl group associated to
an indecomposable symmetrizable generalized Cartan matrix, the group is
infinite if and only if the powers of the element are reduced
words. The latter strengthens known results of Howlett and
Fomin-Zelevinsky. The talk is based on joint work with Helene R. Tyler
and with Allen Pelley.
Mara Neusel, Rings of Invariants Satisfying the Weak
Splitting Principle
Consider a finite group G acting linearly on the ring of polynomial
functions F[V]=F[x_1,...., x_n] in n indeterminants over a field F. The
associated ring of invariants F[V]^G satisfies the weak splitting
principle if there exist finitely many orbits B_1,..., B_l \subseteq V*
of linear forms whose Chern classes generate the ring of invariants. We
present old and new results on this type of invariant rings, and some
open problems.
Ralf
Schiffler, Geometric realizations of
cluster categories
Abstract: The cluster category of a hereditary algebra is
usually defined as a quotient of the bounded derived category of
modules of that algebra. In this talk, we will consider an alternative
definition of the cluster categories of Dynkin types A and D. In type
A, the cluster category is a category of diagonals in a regular
polygon, and, in type D, the cluster category is a category of certain
homotopy classes of paths in a regular polygon with one puncture in its
center.
Jerzy Weyman, Semi-invariants of quivers: new
developments
Abstract: This is joint work with Calin Chindris and Harm Derksen. I
will explain how one can use quiver representations to find
counterexamples to the conjecture of Andrei Okunkov on the
log-concavity of Littlewood-Richardson coefficients.