Noncommutative Day 08*

Wednesday 05 March 2008

Department of Mathematics

Imperial College

Room: Huxley Building

139 from 1.30 to 3.30 p.m., and then room 341 from 4 p.m.

1 .30 p.m. Room 139

Chris Isham (imperial.ac.uk)

TITLE: "Topos theory in the formulation of theories of physics".

ABSTRACT:  "By formulating physics using a topos other than the usual topos of sets, 
quantum theory can be made to 'look like' classical physics. This has striking implications 
for the interpretational problems in quantum theory as well as suggesting new ways 
of constructing 'quantum-like' theories in the context of non-standard models of space and time.  
From a mathematical perspective, the heart of this programme is a type of 'non-commutative spectral theory' 
in which every s.a. operator is represented by an arrow A:\Sigma ---> \cal R where the object $\Sigma$ 
can be viewed as the quantum, topos analogue of a classical state space."

2 .30 p.m. Room 139

Keichi Ito (setsunam.ac.jp)
TITLE: 
ABSTRACT: 

3.30 p.m.

Tea & Coffee

4 p.m. Room 341

Ray Streater (kcl.ac.uk)

TITLE:  A Banach manifold of states for ${\cal B}({\cal H})$. 
ABSTRACT:  We consider ${\cal M}$ as the set of states $\rho$ on ${\cal B}({\cal H})$ 
of the form $\rho=Z^{-1}\exp\{-H\}$, such that $\rho^\beta$ is of trace class 
for some $\beta<1$. We show that $Y_H(X)=1/2[Tr\{\exp\{-H-X\}+Tr\{-H+X\}]-Z$ 
is a quantum Young function for the convex set of $H$-small bilinear forms on 
Dom(H^{1/2}). We topologise a set of states of the form $\rho_X:=Z_X^{-1}\exp\{-H-X\}$ 
for H-small $X$ by using the Luxemburg norm defined by $Y$, to get a neighbourhood $N_H$ of $\rho$. 
We show that any state that lies in both this neighbourhood $N_H$ and a similar one around any 
other point $\rho_X$ in $N_H$ is given equivalent norms using $Y_H$ and $Y_{H+X}$. 
The set of perturbations is thus a coherent topological space, a Banach manifold based on the Orlicz space defined by $Y$. 

5 p.m. Room 341

Vasilli Kolokoltsov (warwick.ac.uk)

TITLE: Boundary value problems for Dirac and Schr\"odinger operators and stochastic calculus (classical and non-commutative)

ABSTRACT:

* Supported by LMS