# Applied Algebraic Topology in Geometry

### University of Aberdeen

Department of Mathematics

## Project Description

In recent years the techniques of algebraic topology have been applied to data analysis with great success, and the resulting subject areas of Applied Algebraic Topology and Topological Data Analysis have seen rapid growth. This project will apply these new techniques to pure mathematics itself, in order to study geometric objects. Topological data analysis studies point clouds, in which each point of the cloud represents a single piece of data. It tries to answer the question “What is the shape of the point cloud?”. It does this by allowing points of the cloud that are close together to merge, and then examining the resulting space. The features of the space that last for a long time as more and more points merge are called the persistent features, and give us a meaningful measurement of the shape of the point cloud. The result is called the Persistent Homology of the point cloud. Topological data analysis has been used to great effect in applications, but has seen little use in pure mathematics itself. This project will seek out applications of topological data analysis in pure mathematics. For example, persistent homology can be applied to the cloud of points in a manifold. In this case, when points that are very close together are merged, the result is an accurate picture of the topology of the manifold. But the persistent features remain to be analysed, and it is hoped that they will encode geometric properties such as volume, diameter and curvature. Applicants for this project should have a good background in pure mathematics, and in particular should have completed first courses in algebraic topology and differential geometry. Persistent homology is a very computational subject, and there are many programming tools for the study of persistent homology. This project offers an opportunity to learn and apply some of these tools, especially in the study of specific examples.

This project can be commenced as a distance learning project.

First courses in Algebraic Topology and Differential Geometry are essential, together with the first degree requirements below.

## Funding Information

This project is advertised in relation to the research areas of the discipline of mathematics. The successful applicant will be expected to provide the funding for Tuition fees, living expenses and maintenance. There is no funding attached to these projects.

## Eligibility Requirements

Applicants should have (or expect to achieve) a UK first class honours degree in Mathematics, or a 2:1 honours degree in Mathematics alongside a Masters with Merit or Distinction, also in Mathematics.

## Application Process

Apply for Degree of Doctor of Philosophy in Mathematics

• State name of the lead supervisor as the Name of Proposed Supervisor
• State ‘Self-funded’ as Intended Source of Funding
• State the exact project title on the application form

When applying please ensure all required documents are attached:

• All degree certificates and transcripts (Undergraduate AND Postgraduate MSc-officially translated into English where necessary)
• Detailed CV

Informal inquiries can be made to Dr Richard Hepworth ([email protected]) with a copy of your curriculum vitae and cover letter. All general enquiries should be directed to the Postgraduate Research School ([email protected])

To apply for this PhD, please use the following application link: https://www.abdn.ac.uk/ncs/departments/mathematics/index.php

##### Overview
• ###### Supervisor Name
Dr R Hepworth & Dr M Grant