# Structure of noncommutative fibre bundles

Name of applicant

Sophie Emma Mikkelsen

Amount

DKK 700,000

Year

2021

Type of grant

Internationalisation Fellowships

## What?

The main object of study in noncommutative geometry is a so-called C*-algebra. Heuristically, this can be viewed as a quantum analogue of a function algebra on a non-existing virtual space. Many concepts from classical geometry, such as compact groups, metric spaces, spin manifolds, vector bundles, K-theory and principal bundles have been successfully carried over to the noncommutative world, leading to interesting mathematical theories with important applications in mathematics and theoretical physics. A concept from geometry which is still not fully understood in the noncommutative setting is that of a fibre bundle. The main goal of the present project is to address this issue by constructing relevant examples and investigating their structure.

## Why?

The proposed project aims to provide us with useful examples and knowledge on the structure of more general noncommutative bundles. Hence it has a great potential to shed new light for future work on obtaining a general definition of noncommutative fibre bundles. Moreover, through a structured understanding of noncommutative bundles we can ask the following: If a C*-algebra can be realised as the total space of a noncommutative bundle, what kind of structure does this impose on it? Answers to this question will provide new ways of extracting information about a given C*-algebra if one can first show that it fits into a noncommutative bundle.

## How?

The Noncommutative Geometry and Topology Group at the host institution Charles University is a young, highly collaborative group undergoing a great expansion. Their research focusses on both quantum algebraic and operator algebraic aspects of noncommutative geometry and topology. The proposed research project fits exactly in the intersection of the groups' research areas. It is therefore my hope to achieve the objectives in the project by a tight collaboration and knowledge exchange.