Projects from Eduard Ortega

I can help with supervision of the following master projects and theses. The projects can be done both individually on in small groups, the level and precise formulation of the problem will depend on the student's background and interests. If you are interested or have an idea outside the scope of these suggestions, please contact me for further discussion.

What is a C*-algebra?

OPERATOR ALGEBRAS

C*-algebras play an important role in functional analysis. They provide the mathematical foundation for quantum physics, and they are important in non-commutative geometry and in the study of singular topological spaces, knot theory, group actions, dynamical systems and quantum groups. However, general C*-algebras are often abstract, counterintuitive, and hard to visualise. Directed, discrete graphs are on the other hand concrete, easy to visualise and can be manipulated pictorially. C*-algebras constructed from such directed, discrete graphs have therefore attracted a great deal of interest because many of the properties of the C*-algebra can be determined by examining the graph it was constructed from.

There are many things a project about graph C*-algebras can deal with, and this can be regulated in accordance with the background of the student and the size of the project. There are for example the basic construction of graph C*-algebras, the gauge invariant uniqueness theorem for graph C*-algebras, Cuntz-Krieger uniqueness theorem for graph C*-algebras, simplicity and the ideal structure of graph C*-algebras, the K-theory of graph C*-algebras and various generalizations of graph C*-algebras such as C*-algebras of topological graphs, Cuntz-Pimsner C*-algebras and C*-algebras of higher rank graphs.

A project in graph C*-algebras doesn’t call for any special background. The courses TMA4145, TMA4230 and MA8107 would be relevant, but are not a necessity. The most important reference is the nice book Graph Algebras (ISBN 0821836609, 9780821836606) by Iain Raeburn, which gives a very nice introduction to the subject and a good overview of it.

The history of associating C*-algebras to dynamical systems is long and successful.

The reason for associating C*-algebras to dynamical systems is, from the point of view of C*-algebra, that one obtain new examples of C*-algebras whose properties can be determined by studying the dynamical systems they were constructed from, and, from the point of view of dynamical systems, that these C*-algebras lead to new invariants of the dynamical systems they were constructed from, and that they are an attractive tool for styding representations of dynamical systems.

The types of dynamical systems to which C*-algebras have been associated have mainly been invertible. There have in recent years been a great interest in associating C*-algebras to non-invertible dynamical systems. Shift spaces (also called subshifts) is a class of non-invertible dynamical systems which is a good place to begin when you want to associate C*-algebras to non-invetible dynamical systems because they are on one hand quite simple, but on the other hand have enough structure to produce interesting C*-algebras.

C*-algebras associated to shift spaces have turned out to be a class of extremely interesting C*-algebras, and has led to some important new invariants of shift spaces.

A project about C*-algerbras associated to shift spaces can be combined in many different ways, and the contents can be regulated in accordance with the background and interest of the student and the size of the project, but could for example include:

1. a short introduction to shift spaces,

2. the basic construction of C*-algebras associated to shift spaces,

3. a description of some of the most important properties of C*-algebras associated to shift spaces,

4. a thorough study of the C*-algerbras associated to different classes of shift spaces such as shift of finite type, sofic shifts and substitutional shift spaces.

A project in this topic doesn’t call for any special background. The courses TMA4145, TMA4230 and MA8107 would be relevant, but are not a necessity.

Literature: A possible starting point could be some lecture notes Toke Meier Carlsen wrote for a summerschool. These can be downloaded from http://arxiv.org/abs/0808.0301.

QUANTUM INFORMATION

Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system is an eigenstate of the observable. Since any two non-commuting observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables

Information is something physical that is encoded in the state of a quantum system. While quantum mechanics deals with examining properties of matter at the microscopic level, quantum information science focuses on extracting information from those properties, and quantum computation manipulates and processes information – performs logical operations – using quantum information processing techniques.

Quantum information theory is an active field of research at the intersection of mathematics, physics, and computer science. In this project, we will focus on the mathematical side of the subject.