Topics in Magnitude

Magnitude homology.
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Magnitude is a mathematical measure of size of 'things' that has arisen from the abstract algebra of category theory but which has connections with classical biological measures of diversity and with classical mathematical notions such as length and volume. Recent work has shown that this magnitude is a shadow of a much richer algebraic structure known as a homology theory. Whilst a fair amount of work has been done on magnitude, very little has so far been done on magnitude homology, although preliminary work of Otter indicates connections with persistent homology which is of use in topological data analysis.

Aims and objectives
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The main aim of this project is a better understanding of magnitude homology.

A significant objective is to compute a variety of examples for magnitude homology. Another objective is to examine the thin literature on magnitude homology and see if conceptual simplifications are possible or indeed which would clarify if this is the 'right' definition of magnitude homology, as recent work of Otter suggests that tweaking the definition leads to connections with persistent homology.

As is typical in pure mathematics, in trying or failing to attain one objective other objectives tend to reveal themselves.

Methodology
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The student will become an expert in the methods of homotopical and homological algebra and of enriched category theory. He will use these methods to compute a variety of examples which can be used to generate conjectures about the general properties of magnitude homology which can then, hopefully be proved.

Alignment
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This research is aligned to algebra, logic and combinatorics, and to geometry and topology and is partially aligned to mathematical analysis and to mathematical biology.