Functional Analysis/Operator Algebras Seminar, Winter 1999

• Date: 1/8/99
  • Speaker: Mihai Putinar, UCSB
  • Title: Introduction to the cohomology of topological algebras I
  • Abstract: We will give an an introduction to the Hochschild cohomology of Frechet modules over Frechet algebras, with emphasis on the first three cohomology groups which have intrinsinc meanings and special interest. We will present various examples mainly from differential and algebraic geometry.

• Date: 1/15/99
  • Speaker: Mihai Putinar, UCSB
  • Title: Introduction to the cohomology of topological algebras II

• Date: 1/22/99
  • Speaker: Bjorn Gustafsson, Royal Institute of Technology, Stockholm
  • Title: Vortex motion in two-dimensional hydrodynamics
  • Abstract: We study the motion of finitely many point vortices in a two-dimensional fluid region of arbitrary geometry. The fluid is supposed incompressible and non-viscous.
    In the case of only one vortex, it interacts with the walls in such a way that it (typically) moves in a periodic manner along a closed curve. For two or more vortices there are interactions also between the vortices, and the motion may be very complicated ("chaotic"). The system may even break down in a finite time due to collisions. An explicit example of a collosion between three vortices was given by Y. Kimura in 1988.
    However, Martin Flucher and myself have shown a couple of years ago that, in the case of just two vortices, collisions can never occur. The proof of this is based on a detailed study of the Green's function of the domain.

• Date: 1/29/99
  • no meeting

• Date: 2/5/99
  • Speaker: Mihai Putinar, UCSB
  • Title: Introduction to the cohomology of topological algebras III

• Date: 2/12/99
  • Speaker: Anne Louise Svendsen
  • Title: On the notion of amenability for groups

• Date: 2/19/99
  • Speaker: Bernie Russo, UC Irvine
  • Title: Jordan approach to representations of the Lorentz group
  • Abstract: The Canonical Anticommutation Relations are formulated in the context of Banach Jordan triple systems. They are used to construct in an elementary way a spin factor structure on finite dimensional complex Euclidean space and a subgroup of the corresponding linear isometry group. Some representations of the rotation and Lorentz group which are analogous to spinor representations are realized in this group and its Lie algebra.

• Date: 2/26/99
  • no meeting

• Date: 3/5/99
  • Speaker: Dietmar Bisch
  • Title: Planar algebras and subfactors
  • Abstract: Subfactors with finite Jones index have an amazingly rich mathematical structure and an interplay of analytical, algebraic-combinatorial and even topological techniques is intrinsic to the theory. A subfactor can be viewed as a group-like object that encodes what one might call generalized symmetries of the mathematical or physical situation from which it was constructed. To decode this information one needs to compute a system of inclusions of certain finite dimensional C$^*$-algebras naturally associated to the subfactor. This structure can be described as a planar algebra, that is a graded vector space whose elements are represented by labelled n-boxes which can be combined in planar, but otherwise quite arbitrary ways. This allows one to employ a diagrammatic language to manipulate certain operators on a Hilbert space associated to a subfactor and to carry out intricate calculations using topological arguments.
    
    We will discuss the notion of planar algebras and present some examples such as the Temperley-Lieb and Fuss-Catalan algebras.