I am a mathematician with research interests in algebraic topology, homological algebra and category theory. I am currently a teaching fellow at the University of Leeds.

Research

Here a list of my publications and preprints. This information is also available at my google scholar page and my ORCiD page 0000-0003-1136-9295.

Papers

1. PROPs for involutive monoids and involutive bimonoids, Theory and Applications of Categories, Vol. 35, 2020, No.42, pp 1564-1575. Journal, arXiv.
2. Hyperoctahedral homology for involutive algebras, Homology, Homotopy and Applications, Vol. 24 (1), 2022, pp 1-26. Journal, arXiv.
3. Composing PROBs, Theory and Applications of Categories, Vol. 38, 2022, No. 26, pp 1050-1061. Journal, arXiv.
4. E-infinity structure in hyperoctahedral homology, Homology, Homotopy and Applications, Vol. 25 (1), 2023, pp 1-19. Journal, arXiv.
5. Reflexive homology, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2024. Journal, arXiv.
6. Categorifying equivariant monoids, Proceedings of the American Mathematical Society, Volume 152, Pages 3689-3704, 2024. Journal, arXiv.
7. Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories. arXiv. To appear in Comptes Rendus Mathématique.

Preprints

1. A comparison map for symmetric homology and gamma homology. arXiv.
2. Generalized rook-Brauer algebras and their homology. arXiv.
3. Cohomology of Tanabe algebras. Joint work with Andrew Fisher. arXiv.
4. Cohomology of coloured partition algebras. Joint work with James Cranch. arXiv.

Thesis

Functor homology for augmented algebras. Available on the White Rose eTheses website here. Supervised by Sarah Whitehouse.

Teaching

University of Leeds (Lifelong Learning Centre, 2023-)

• LLLC0203: Introduction to Physics. Lecturer.
• LLLC0197: Introduction to Mathematics. Tutorial demonstrator.
• LLLC0185: Waves and Particles. Lecturer.
• LLLC0202: Electromagnetism and Thermal Physics. Lecturer.
• LLLC0198: Introduction to Pure Mathematics. Tutorial demonstrator.

University of Leeds (School of Mathematics, 2022-2023)

• MATH3225: Topology. Module leader.
• MATH1060: Introductory Linear Algebra. Module leader.
• MATH3017: Calculus in the complex plane. Module leader.
• MATH1026: Sets, Sequences and Series. Tutorial demonstrator and marker.
• MATH1005: Core Mathematics. Tutorial demonstrator and marker.
• MATH1225: Introduction to Geometry. Tutorial demonstrator and marker.
• MATH1515: Interest Rates and Cashflow Modelling. Tutorial demonstrator, coursework marker and exam marker.
• MATH3001: Project supervisor. I supervised a group of third year undergraduates investigating the Fibonacci numbers.
• I have been a mentor in a scheme designed to help PhD students who are new to teaching.

University of Sheffield (SoMaS 2015-2022)

Here is a list of my teaching at the University of Sheffield, in no particular order. The classes in list come in a range of flavours: lectures, problems classes, flipped learning classes, outreach classes and online assessment.

• Lecturer for MAS003: Foundation mathematics. As part of my role as lecturer I developed a new set of lecture notes for part of the course as well as being responsible for the online tests using STACK. I have also acted as tutorial demonstrator for this module.
• Lecturer for MAS248. This is a mathematics module for second year undergraduates in chemical engineering.
• MAS221: Analysis. I have been a tutorial demonstrator, marker and exam marker for this module. I have also delivered some lectures as cover. I have written and coded online tests for this module in both AiM (Assessment in Mathematics) and STACK.
• MAS220: Algebra. Tutorial demonstrator and marker. I have also written and coded online tests in AiM for this module.
• MAS114: Numbers and Groups. Tutorial demonstrator and marker.
• MAS241 tutorial demonstrator. This is a second year mathematics module for second year undergraduates in electrical engineering.
• MAS279: Career Development Skills. Marker for reflective work.
• Year 12 and Year 13 A-level enrichment classes and STEP classes.
• Dissertation Support for M.Sc. Statistics students.

Conferences and research talks

Organized

I have co-organized two early career topology research conferences at the University of Sheffield:

1. EL:ECTRIC in 2017 with Gemma Halliwell, details of which can be found here.
2. ECTR in 2019 with Jordan Williamson, details of which can be found here.
3. In July 2024, Joao Faria Martins and I organized a meeting of the TTT in Leeds (TTT120).

Attended

Here is a list of conferences I have attended:

• Transpennine Topology Triangle meetings, 2015 onwards.
• ECSTATIC, Imperial College, 2016.
• Young Topologists Meeting, Copenhagen 2016 and Stockholm 2017.
• US Talbot workshop on obstructruction theory for structured ring spectra, Gooding, Idaho, 2017.
• British Topology Meeting, Milton Keynes 2018, Warwick 2019, Durham 2022, Sheffield 2023, Aberdeen 2024.
• Homotopy Harnessing Higher Structures, Isaac Newton Institute, Cambridge, 2018.
• OATS (Online algebraic topology seminar), 2020.
• Higher Homotopical Structures opening workshop (online), CRM, Barcelona, 2021.
• TopFlavours, Warwick, 2021.
• A Festival Remembering Victor Snaith: Topology, Number Theory and Interactions, 2023.

Research talks

• Functor homology, ECSTATIC, Imperial College, 2016.
• Now that’s what I call…homology theories for algebras!, University of Sheffield topology seminar, 2019.
• Comparing symmetric homology and gamma homology, Transpennine Topology Triangle, Warwick, 2020.
• Distributive laws, double categories and PROPs for involutive bimonoids, MiaowMiaow online double category seminar, 2021.
• Hyperoctahedral homology, TopFlavours online, Warwick, 2021.
• What’s the hype about hyperoctahedral homology?, British Topology Meeting, Durham, 2022.
• A talk on the PROBlem of PROducing PROPer indexing categories for categories of monoids, University of Sheffield topology seminar, 2022.
• Categorifying equivariant monoids, Technische Universität Dresden, 2023.
• Reflections on reflexive homology, British Topology Meeting, Sheffield, 2023.
• Homology of generalized rook-Brauer algebras, University of Sheffield topology seminar, 2023.
• The homology of diagram algebras, Pure Maths Colloquium, University of Sheffield, 2023.
• A survey of crossed simplicial groups and their homology theories, Algebraic Topology seminar, Universität Hamburg, 2024.
• Reflexive homology and beyond, Topology seminar, University of Warwick, 2024.
• Cohomology of diagram algebras, British Topology Meeting, Aberdeen, 2024.
• Groups with an oriented parity sign (or, why owning a thesaurus shouldn’t be a prerequisite for studying equivariant Hochschild homology), University of Sheffield topology seminar, 2024.

Service Work

• Referee for European Journal of Mathematics and Journal of Pure and Applied Algebra.
• Reviewer for zbMath Open.
• Reviewer for Mathematical Reviews (MathSciNet).
• I created and maintained (2017-2020) the Sheffield Algebraic Topology website.

Outreach activities

• Student assistant at Sheffield PopMaths quiz 2016, 2017 and 2019.
• Team member for Year 12 STEM fair, 2017.
• Open Day stall team member, 2018 and 2019.
• Submitted puzzles for BBC4’s Today programme, which were used on air and in the subsequent book, 2018.
• Discovery Night stall team member for Key Stage 2 students, 2019.
• UKMT Maths Olympiad marking, 2019.

Seminars and reading groups

I enjoy taking part in seminars and reading groups. Here are some that I have been involved in.

Organized

• Throughout 2017 and 2018 I co-organized the University of Sheffield Student Seminar series with Jake Shipley.
• In 2019, I co-organized a reading group on Topological Hochschild Homology with James Brotherston.

Talks given in reading groups

• Phylogenetic trees and cancer evolution, Applied Topology reading group 2015.
• Representations of quivers and Gabriel’s theorem, Pure maths postgraduate seminar, 2015.
• Introduction to vector bundles, Topological K-theory reading group, 2016.
• Knots and DNA, Student seminar, 2016.
• Formal group laws and Lazard’s theorem, Chromatic homotopy theory reading group, 2017.
• A simplicial view of algebraic topology, Pure maths postgraduate seminar, 2017.
• Hochschild homology of enriched categories, Magnitude homology reading group, 2018.
• Magnitude homology is Hochschild homology, Magnitude homology reading group, 2018.
• Machine learning and the World Cup, Student seminar, 2018.
• Hochschild, Shukla and cyclic homology, Topological Hochschild homology reading group, 2019.
• Online tests, Teaching and Learning seminar, 2019.
• Morita equivalences, Derived commutative algebra reading group, 2019.
• The cacti operad, String topology reading group, 2020.
• All Q’d up, Condensed maths reading group, 2022.
• Spaces, condensed sets and condensed abelian groups, Condensed maths reading group, 2022.
• From Algebraic Topology to Terrapin Station in three easy steps, Sheaf seminar, Sheffield, 2022.
• Just another whistlestop (of stable homotopy theory), Telescope conjecture reading group, 2023.
• The most tip top…TC!, Telescope conjecture reading group, 2023.