Til bevillingsoversigt

Ergodic theory and the structure of KMS weights

Carlsbergfondets internationaliseringsstipendier

What

The purpose of my project is to build a new bridge between two areas of Mathematics. On one side, we have operator algebras, an important research area in Mathematics pioneered in the 1930’s to provide mathematical models for quantum statistical mechanical systems. On the other side, we have ergodic theory, a classical and beautiful area of Mathematics concerned with studying certain properties of groups acting on spaces.

In my project, I will study a class of examples that allows me to use ideas from ergodic theory to understand the objects used to model equilibrium phenomena in operator algebras, called KMS weights. The aim is to use this bridge to deepen our understanding of these KMS weights and therefore our mathematical understanding of these equilibrium phenomena.

Why

The class of examples I will consider in this project are very fundamental in the theory of operator algebras, but so far, the question of describing their KMS weights has been completely overlooked in literature. A better understanding of the KMS weights for these examples will therefore be an important contribution in its own right.

If one takes a helicopter view of the research area of describing KMS weights, there are some big important problems regarding KMS weights that we currently cannot solve. My project contains some of these problems in a scaled down version, so a successful outcome of my project might also move us a step forward with answering these bigger questions.

How

Even though one can do excellent research in mathematics with no more equipment than pencils, paper and a decent library, it is often of vital importance to discuss problems with experts. To achieve the expected outcome of this project I will travel to Leuven in Belgium where one of the world’s leading experts in operator algebras and KMS weights are working. With his supervision, and as part of the excellent research group in Leuven, I have the optimal conditions to carry out my project.

SSR

The project described in this application is basic research and it does not have any immediate impact on society. However, in the long run this area of research will give us better mathematical models of certain equilibrium phenomena in Physics.