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The University of Southampton
Mathematical Sciences

Research Group: Topological Data Analysis

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The modern world thrives on data. While we still tend to think of data as numbers, most of the data created in modern science, medicine, biology, in digital economy, in commerce, digital security is vastly more varied: it can be a set of images, a collection of objects, a dynamic profile of an evolving system, sound or video recordings. A common thread in modern data science is the analysis, classification and comparison of complex shapes. This is why topology, a branch of mathematics dedicated to the study of shape, is so useful in this context. Topological data analysis has been developed to provide numerical characteristics that allow the classification and comparison of complex structures. Its flexibility, computational power, and impressive visualisation tools have been used in recent breakthrough results in the study of cancer, asthma, chronic lung diseases, and many others. This subject offers challenging theoretical problems matched with opportunities for important applications and is perfect for an enterprising PhD student.

We are interested in the following specific topics:

  • Theoretical foundations of multiparameter persistent homology: Persistent homology has emerged as a key computational tool in topological data analysis, where it provides numerical characteristics of the shape of data at a range of scales. It is also a very interesting new homology theory, with many possible applications within mathematics. Our group is studying the formal properties of this new theory, its links to other homology theories and other structures in algebraic topology. We are in particular investigating symmetries, group actions, and localisation. A significant part of our work is devoted to a deeper understanding of stability (and instability) of persistent homology.
  • Mapper and clustering methodologies: Another important tool in TDA is mapper, first proposed by Carlsson, Singh, and Zomorodian. Mapper can be regarded as a form of flexible hierarchical clustering, which combines combinatorial information of the data with its statistical properties. We are also interested in the stability properties of mapper and its interactions with other topological methods.
  • Topology and neural networks: Neural networks are a very successful part of Machine Learning and a central methodology in the development of Artificial Intelligence (AI). The theoretical foundations of this methodology are not so well understood and again topology can be of great help here. We are developing topological methods to understand topological complexity of the data which will help suggest the correct architecture of the network.
  • Geometry of synchronisation: There are important examples of data which is presented in the form of a set of objects, like CT scans, collections of bones, molecules, etc. We have developed a new methodology that combines discrete Hodge theory of the graph Laplacian with ideas regarding synchronisation, developed by Singer and coworkers. We are investigating higher dimensional extensions of these ideas and their applications in data science.
  • Combinatorial methods in topological data analysis: While much emphasis has been put on the computational aspects of TDA, there is still much to do in terms of the development of the foundations. In particular, many of the powerful combinatoric methods developed in algebraic topology (in particular in relation to toric topology, homotopy theory, group actions, etc) have not yet found a translation into the topological data analysis. We are actively developing this area.

This is not a complete list of our interests, and we always welcome new collaborations. If you would like to work with us, or study for a PhD, then please contact us.

Related Research Centres

The group welcomes applications for postgraduate studies. Please contact group members (email, telephone, or in person) for more information; and when ready, please apply through the Graduate School application page.

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