1 2 M ay 2 00 3 Measurable Categories and 2 - Groups

2 3 M ay 2 00 7 From triangulated categories to abelian categories – cluster tilting in a general framework

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal oneorthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

ar X iv : m at h / 06 05 27 6 v 2 [ m at h . D S ] 1 1 M ay 2 00 6 SUPERRIGIDITY , WEYL GROUPS , AND ACTIONS ON THE CIRCLE

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with Homeo+(S ) as the target group. We are motivated by Ghys’ theorem stating that any representation ̺ : Γ → Homeo+(S ) of an irreducible lattice Γ in a semi-simple real Lie group G of higher rank, either has a finite orbit or, up to a semi-conjugacy, extends to G which act...

1 1 M ay 2 00 7 WEIGHT 2 BLOCKS OF GENERAL LINEAR GROUPS AND MODULAR ALVIS - CURTIS DUALITY

We obtain the structure of weight 2 blocks and [2 : 1]-pairs of q-Schur algebras, and compute explicitly the modular Alvis-Curtis duality for weight 2 blocks of finite general linear groups in non-defining characteristic.

1 M ay 2 00 7 BOUNDARY CROSS THEOREM IN DIMENSION 1 WITH SINGULARITIES

Let D and G be copies of the open unit disc in C, let A (resp. B) be a measurable subset of ∂D (resp. ∂G), let W be the 2-fold cross ( (D ∪A)×B ) ∪ ( A× (B ∪G) ) , and let M be a relatively closed subset of W. Suppose in addition that A and B are of positive one-dimensional Lebesgue measure and that M is fiberwise polar (resp. fiberwise discrete) and that M ∩(A×B) = ∅. We determine the “envelop...

1 9 M ay 2 00 5 Lefschetz formulae for p - adic groups