Noncommutative Day 06

Thursday 07 December 2006

Department of Mathematics

Imperial College

Room: Huxley Building 342

Time: 14.00- 18.30

J. Brodzki (Southhampton)

Title:Topological invariants of noncommutative spaces.

Abstract: In noncommutative geometry, operator algebras are used to capture topological information about noncommutative spaces. Interesting examples arise when geometry interacts with analysis, for example, in analysis and geometry of groups or mathematical physics. In this talk we shall consider two sources of such examples. In group theory, the reduced C*-algebra provides topological information about spaces of representations of the group. On the other hand, one of the important points in Connes's characterisation of noncommutative manifolds is Poincare duality, which is implemented using a fundamental class in bivariant K-theory. We shall aim to demonstrate how the general philosophy of noncommutative geometry works in practice.

I.Krolak (Wroclaw)

Title: Ornstein-Uhlenbeck semigroup for general commutation relations.

Abstract: We study a certain class of von Neumann algebras generated by selfadjoint elements \omega _i= a_i + a_i⁺ , for a_i and a_i⁺ satisfying the general commutation relations i.e.

a_i a_j ⁺ = ∑_r,s t ^ir _js a_r⁺a_s + \delta _ij Id.

We assume that operator T for which the constants t ^ir _js are matrix coefficients satisfies the braid relation. This algebras are type II_1 factors and therefore we can fairy easy define non-commutative L^p spaces. Using second quantization we construct generalized Ornstein-Uhlenbeck semigroup and analyze its contractivity properties as a map between these non-commutative L^p spaces.

D. Petz (Budapest)

Title: Noncommutative probability

Abstract: In an operator algebraic approach, an overview will be given about noncommutative integration, conditional expectation and independence. Central limit theorems in the framework of free probability and canonical commutation relation will be discussed.

Ville Turunen (Helsinki University of Technology, Finland).

Title : Pseudo-differential operators on Lie groups

Abstract: "On a Lie group, we present a pseudodifferential operator by a family of convolution operators (i.e. by an operator valued mapping on the group). This family is a natural analogue of the symbol of a pseudodifferential operator. We discuss this approach demonstrating its relation to pseudodifferential calculus on Euclidean spaces."

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PDE Working Group

Friday 08 December

Room: Huxley Building 658

D. Petz (Budapest)

Tile: The flavour of noncommuative integration

Abstract: The traditional and noncommutative integration will be compared, the noncommutative L^p norms will be discussed. Among the applications the mathematical foundations of quantum theory will be sketched.