The workshop will consist of three lecture series of 4-5 hours each.

We will introduce some of the main concepts in local representation theory of finite groups: Brauer characters, decomposition numbers, Cartan invariants, lifting of idempotents, blocks, defect groups, heights and defects of characters. Trivial source modules and the Brauer homomorphism will be presented, and Brauer's three main theorems in block theory will be stated. If time permits, some of the main open problems in block theory will also be addressed, such as Donovan's Conjecture, and some of the recent progress on this conjecture will be mentioned.

The course will develop the block theoretic background material required to recast the most prominent conjectures in block theory in terms of statements relating the algebra structure of a block of a finite group to the fusion system of the block. We will review the classic theory of Brauer pairs due to Alperin and Broue, leading to Puig's notion of fusion systems associated with a block. We will relate this to another block theoretic key concept, the notion of a source algebra of a block, and describe in what way source algebras are an indispensable tool in block theory.

The lectures will give an introduction to Lusztig's theory, where methods from algebraic geometry are used to determine the irreducible characters of finite reductive groups. This will include a discussion of characters of groups with a non-connected center and, if time permits, also disconnected groups.

*Burkhard Külshammer*, Friedrich Schiller Universität Jena__Basic local representation theory__We will introduce some of the main concepts in local representation theory of finite groups: Brauer characters, decomposition numbers, Cartan invariants, lifting of idempotents, blocks, defect groups, heights and defects of characters. Trivial source modules and the Brauer homomorphism will be presented, and Brauer's three main theorems in block theory will be stated. If time permits, some of the main open problems in block theory will also be addressed, such as Donovan's Conjecture, and some of the recent progress on this conjecture will be mentioned.

*Markus Linckelmann*, City University London**Block theory and fusion systems**The course will develop the block theoretic background material required to recast the most prominent conjectures in block theory in terms of statements relating the algebra structure of a block of a finite group to the fusion system of the block. We will review the classic theory of Brauer pairs due to Alperin and Broue, leading to Puig's notion of fusion systems associated with a block. We will relate this to another block theoretic key concept, the notion of a source algebra of a block, and describe in what way source algebras are an indispensable tool in block theory.

*Meinolf Geck*, Universität Stuttgart**The representation theory of finite reductive groups**The lectures will give an introduction to Lusztig's theory, where methods from algebraic geometry are used to determine the irreducible characters of finite reductive groups. This will include a discussion of characters of groups with a non-connected center and, if time permits, also disconnected groups.