Activities This Week

The stable module category is obtained from the category of modules over a ring by factoring out the projective modules. In this setting, the syzygy of a module becomes a well defined functor, so for instance, the classical Hilbert’s syzygy theorem, can be stated as saying this functor is nilpotent. In this talk we present some new properties of the syzygy functor over a commutative noetherian ring. We then explain how to associate to the stable category a stabilization, obtaining the singularity category of a ring (or a scheme). Finally, we explain how relations between the stable category and the singularity category are related to some homological conjectures in noncommutative algebra.

Given a non elementary locally compact hyperbolic group G equipped with a left invariant metric d one can define a measure on the Gromov boundary called the Patterson Sullivan measure associated to d. This measure is non singular with respect to the G action and contains geometric information on the metric. I will discuss the koopman representations of these actions and sketch a proof of their irreducibility and classification (up to unitary equivalence), generalizing works of Garncarek in the discrete case. I will also describe connections with a recent work of Caprace, Kalantar and Monod on the type I property for hyperbolic groups.