[Image] Emily Roff

I am a PhD student at the University of Edinburgh, working with Tom Leinster. I'm a member of the Geometry and Topology group in the Hodge Institute, and my primary research interests are in category theory and its applications.

Before joining the University of Edinburgh in September 2018 I took the MASt in Pure Mathematics at the University of Cambridge, where I wrote my Part III essay about commuting algebraic operations, under the supervision of Martin Hyland. Before that, I studied for a BSc with the Open University and King's College London.

On this page: Writing | Talks | Conferences and travel | Teaching | Outreach and exchange | Past lives | Contact


The maximum entropy of a metric space (with Tom Leinster), arXiv:1908.11184

Abstract We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Rényi entropies of information theory.

We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X.

Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.


Jun 2020 Magnitude homology, from scratch Pure PGR Seminar, Southampton (via Teams)
Dec 2019 Diversity and magnitude LMS Applied Algebra and Geometry research network, Oxford
Nov 2019 Hochschild homology for enriched categories Hodge Club, Edinburgh
Jul 2019 The maximum diversity of a compact metric space Magnitude 2019, ICMS, Edinburgh
Apr 2019 The magnitude and diversity of a metric space TopApp workshop, IST Austria
Apr 2019 The magnitude of an enriched category GEARS seminar, Edinburgh/Glasgow
Mar 2019 Diversity: developments in the biology of mathematics Hodge Club, Edinburgh
Jan 2019 Building complexes from point clouds Persistent homology working seminar, Edinburgh
Mar 2018 Commutative monads Part III seminar, Cambridge

Conferences and travel

Jul 2020 Applied Category Theory 2020, MIT (remote conference)
Mar 2020 Organizer of Topics in Category Theory: A Spring School, Edinburgh (programme, photo)
Dec 2019 LMS Applied Algebra and Geometry research network, Oxford
Jul 2019 Category Theory 2019, Edinburgh
Jul 2019 Magnitude 2019: Analysis, Category Theory, Applications, Edinburgh
May 2019 PSSL 105, Palermo
May 2019 Visiting Universitat Autònoma de Barcelona
Apr 2019 TopApp workshop on applied and computational topology, IST Austria
Oct 2018 The Philosophy and Physics of Noether's Theorems, London


As a graduate teaching assistant in the School of Mathematics at Edinburgh, I have tutored the following courses.

Spring 2020 Honours Algebra
Autumn 2019 Advanced Mathematical Economics (in the School of Economics)
Autumn 2019 Accelerated Proofs and Problem Solving
Autumn 2019 Honours Analysis Skills
Spring 2019 Honours Algebra Skills

Outreach and exchange

Sep 2019 — Performing Arts Forum, Saint-Erme-Outre-et-Ramecourt
Instructor at Mobilis in Mobili (led by Gabriel Catren)—a week-long workshop, addressed to non-scientists, introducing the history that led from the first axiomatic formulation of geometry in Euclid's Elements to Einstein's theory of special and general relativity. Presented a class on Euclidean and Cartesian geometry, and a workshop aimed at "discovering" Gaussian curvature and the Theorema Egregium.
Nov 2018, May 2019 — Robert Schumann Hochschule, Düsseldorf
Invited participant at Form and Formalism (led by Julian Rohrhuber and Tzu Chien Tho)—a seminar for students of music informatics and media theory with an interest in abstract algebra and category theory. Led discussions about universal properties, diagrams and limits.
Jun 2018 — Performing Arts Forum, Saint-Erme-Outre-et-Ramecourt
Instructor at Categories for the Hard-Working Non-Mathematician (led by Gabriel Catren)—a week-long, philosophically oriented introduction to the basics of category theory for people without a university background in mathematics. Presented three two-hour lectures: on sets, equivalence relations and groupoids; on categories and functors; and on algebraic methods in topology.
Feb 2018 — Skowhegan Gallery and NYU, New York
Invited participant at the Institute of Physical Sociality—a research group interested in mathematics and physics, ethics, gender and race, organized by Arika and led by Karen Barad, Gabriel Catren, Fred Moten and Fernando Zalamea.

Past lives

From 2011 through 2016 I worked as a welfare rights adviser with Child Poverty Action Group, Drumchapel Money Advice Centre, and WestGAP. Before that, I was a producer of music and arts projects with Arika and Tracer Trails. I have an undergraduate degree from the School of Scottish Studies at the University of Edinburgh, which involved training in anthropological theory and ethnographic research methods.

As my outreach and exchange activities suggest, I'm interested in talking with non-mathematicians about what mathematics has to do with understanding the social world, and I'm a fan of pedagogical experiments that aim to make such conversations possible.

Contact details

Office JCMB 5621

Email firstname.lastname@ed.ac.uk

This page was last updated on 2 June 2020. Photo by Christian Töpfner.