Workshop

"Perspectives in Representation Theory"

Physikzentrum Bad Honnef (July 7-9, 2006)

Michel Broué (Paris): Representation Theory of Finite Groups

Abstract: Connection with local theory of finite groups and presentation of some of the main conjectures (Alperin's Weight Conjecture and some of its variants and extensions, Abelian Defect Group Conjecture).

Gerhard Hiss (Aachen): Representations of finite groups of Lie type in non-defining characteristics

Abstract: My talk describes the aims, methods and the state of the art in the representation theory of finite groups of Lie type. Connections to other branches of representation theory such as to representations of Hecke algebras, q-Schur algebras or quantum  groups will be explained.

Alan Huckleberry (Bochum): Actions of noncompact real forms in complex geometry

Abstract: Beginning with a holomorphic or algebraic action of a complex reductive group on a complex variety, we consider the restriction of the action to a noncompact real form in situations where associated momentum geometry defines complex geometric objects of interest. Recent results in certain restricted settings which are relevant for the realization of the representation theory of the real form will be discussed.  These include precise descriptions of cycle spaces and related harmonic analysis. Emphasis will be placed on open problems and potential interaction with various areas of representation theory.

Jens Carsten Jantzen (Aarhus): Representations of reductive groups and their Lie algebras: Old problems and new methods

Bernhard Keller (Paris): Cluster algebras and representations of quivers

Abstract: Cluster algebras were invented by S. Fomin and A. Zelevinsky in spring 2000 as a tool to approach Lusztig's theory of canonical bases in quantum groups and total positivity in algebraic groups. Since then, cluster algebras have become the center of a rapidly developing theory, which has turned out to be closely related to a large spectrum of other subjects, notably Lie theory, Poisson geometry, Teichmüller theory and, last not least, quiver representations. In this talk, we will introduce cluster algebras, illustrate the notion on a number of examples and present a construction by Caldero-Chapoton which yields a surprisingly direct connection with representations of quivers.

Toshiyuki Kobayashi (Kyoto): Branching problems of unitary representations

Abstract:The branching problem asks how a given irreducible representation of a group decomposes when restricted to a subgroup. Making an observation on wild features of branching problems of unitary representations for non-compact subgroups, we address the question of finding a criterion for discreteness of spectrum and finiteness of multiplicities in the irreducible decomposition of the restriction of reductive Lie groups. Applications to other areas of mathematics such as topology of locally symmetric spaces, and global analysis on indefinite-Riemannian homogeneous spaces will be illustrated.

Markus Reineke (Münster): Moduli of representations

Abstract: The geometric study of moduli spaces of representations, i.e. varieties parametrizing certain classes of representations of algebras up to isomorphism, is a promising road towards understanding wild classification problems in representation theory. The talk will focus on moduli of representations of quivers. Some recent results, mainly concerning the computation of geometric and topological invariants, will be stated. These results, and the methods applied to derive them, point towards interactions to several research areas in Algebraic Geometry and Noncommutative Algebra. Several open problems and possible research directions in connection to these areas will be formulated.

Idun Reiten (Trondheim): Cluster categories

Abstract: In this lecture we give an introduction to cluster categories, which were inspired by the theory of cluster algebras developed by Fomin-Zelevinsky. We discuss how they give a natural framework for modelling the main ingredients involved in the definition of a cluster algebra, in the acyclic case with no coefficients.

Wolfgang Soergel (Freiburg): Geometric and combinatorial aspects of representation theory

Abstract: We discuss mainly connections and possible interactions between different branches of modern representation theory. We speculate about promising directions for future research, again emphasizing the potential of cooperation with people not belonging to our field.