Daniel Heath
Personal Webpage
Alan Turing Building
Manchester
M13 9PY
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Daniel Heath
Personal Webpage
Alan Turing Building
Manchester
M13 9PY
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About Me
I am a final year PhD Mathematics student at the University of Manchester, supervised by Prof. Mark Kambites and Dr. Nóra Szakács. I previously completed my MMath programme at the University of Manchester from 2017-2021. My two Master's projects were Introduction to Semigroup Theory and The Role of Fractal Geometry in Diophantine Approximation, supervised by Prof. Mark Kambites and Dr. Faustin Adiceam respectively.
My pronouns are currently He/Him.
My research lies primarily in Semigroup Theory, particularly geometric areas, but I can also be found working with graph theory, combinatorics, geometric (semi)group theory, cancellative and inverse monoids with an eye to C*-algebras, and ample groupoids. My (tentative) thesis title is Left Adequate Expansions of Right Cancellative Semigroups.
When not staring at graphs, I am a dysgwyr Cymraeg, hobbyist programmer, nonogrammer, variant sudokuer and table tennis player-coach.
I am also an amateur cruciverbalist and have made a range of Cryptic Crosswords which you are free to have a look at.
I'm always excited to hear about interesting maths, especially semigroups. Feel free to contact me!
Publications and Preprints
Preprints should be cited with caution. Please contact me if you'd like to discuss further.[2] A collection of cancellative, right LCM, not group-embeddable monoids
(with Milo Edwardes)
May 2024, arXiv:2405.20197. Submitted.
Abstract: By classical results of Malcev, cancellative monoids need not be group-embeddable. In this paper, we describe and give presentations for and study an infinite family Mn of cancellative monoids which are not group-embeddable, originating from Malcev's original work. We show that Mn is right LCM for n ≥ 2, owing to applications in the study of C*-algebras by Brix, Bruce and Dor-On. We finish by showing that M1 is not right LCM, but is 2-aligned.
[1] Pretzel monoids
(with Mark Kambites and Nóra Szakács)
May 2024, arXiv:2405.00589. Submitted.
Abstract: We introduce an interesting class of left adequate monoids which we call pretzel monoids. These, on the one hand, are monoids of birooted graphs with respect to a natural `glue-and-fold' operation, and on the other hand, are shown to be defined in the category of left adequate monoids by a natural class of presentations. They are also shown to be the free idempotent-pure expansions of right cancellative monoids, making them, in some sense, the left adequate analogues of Margolis-Meakin expansions for inverse monoids. The construction recovers the second author's geometric model of free left adequate monoids when the right cancellative monoid is free.
Where you may have seen me...
Teaching and External Mathematics
2023--Pres. Coordinator for MathsBombe.
2022--2024 MATH20621 Programming with Python.
2022--2023 MATH20201 Algebraic Structures 1.
I've also performed marking, covering and invigilation duties for various MATH units.
Organisation
Winter 2024 Reading Group on C*-Algebras, University of Manchester.
2023/2024 Pure Postgraduate Seminar, University of Manchester.
With Anton Farmar and Peej Ingarfield.
25-03-2022 NBSAN XXXI: Young NBSAN, University of Manchester.
With Thomas Aird, Dmitry Kudryavtsev, Carl-Fredrik Nyberg Brodda and Georgia Schneider.
Talks and Seminars
(11-12-2024) TBA, York Semigroup, University of York.
25-10-2024 Graph Expansions of Monoids and Groups, Semigroup Seminar, University of Manchester.
27-09-2024 Three Cornerstones of Pure Mathematics, MRSC XVIII, University of Manchester.
04-09-2024 Pretzel Monoids: A Dive into Geometric Semigroup Theory, HDP 1-Day at Heilbronn Annual Conference 2024, University of Bristol. [Slides]
02-07-2024 Pretzel Monoids: Left Adequate Expansions of R. Cancellative Monoids, Theoretical and Computational Alg. '24, University of Aveiro, PT. [Slides]
19-06-2024 Pretzel Monoids: Left Adequate Expansions of R. Cancellative Monoids, 75th British Mathematical Colloquium, University of Manchester. [Slides]
19-06-2024 Pretzel Monoids, 75th British Mathematical Colloquium, University of Manchester. [Poster]
13-06-2024 Non-Embeddable Right LCM Semigroups, UK Operator Algebras Conference, Newcastle University. [Slides]
07-06-2024 C*-Algebras, Right LCM and Interval Monoids, Semigroup Seminar, University of Manchester.
15-04-2024 Embedding (Right LCM) Semigroups into Groups, Student Algebra Seminar, University of Manchester.
12-04-2024 Embedding (Right LCM) Semigroups into Groups, Semigroup Seminar, University of Manchester.
21-02-2024 Pretzel Monoids: A Dive into Geometric Semigroup Theory, Introduction to Modern Advances in Algebra, University of Exeter. [Slides]
16-02-2024 Pretzel Monoids: A Dive into Geometric Semigroup Theory, Pure Postgraduate Seminar, University of Manchester. [Slides]
02-02-2024 Non-Embeddability of Malcev Semigroups, Pure Postgraduate Seminar, University of Manchester.
26-01-2024 Monogenic Free Adequate Semigroups, Semigroup Seminar, University of Manchester.
31-10-2023 On Cayley Graphs of Algebraic Structures, Student Algebra Seminar, University of Manchester.
29-09-2023 Left Adequate Semigroups and Graph Expansions, MRSC XVII, University of Manchester.
28-06-2023 Idempath Identifications, Semigroup Seminar (Away Day), Unversity of Manchester.
08-06-2023 Unary Monoids: Inverse and Adequate, Student Algebra Seminar, University of Manchester.
19-05-2023 Ehresmann Monoids, Semigroup Seminar, University of Manchester.
02-12-2022 Properness for Adequate Monoids, Pure Postgraduate Seminar, University of Manchester.
18-11-2022 Adequate and Ample Monoids, Semigroup Seminar, University of Manchester.
11-08-2022 Food Supply Graphs, HDP Summer Programme 2022, ICMS Edinburgh.
17-12-2021 Free Adequate Semigroups, Semigroup Seminar, University of Manchester.
Selected other Conferences and Meetings
Weekly 23/24 Pure Postgraduate Seminar, University of Manchester (Organiser).
24-07-2023 HDP Summer Programme 2023, University of Bristol.
03-07-2023 Theoretical and Computational Algebra 2023, Pocinho, PT.
24-06-2022 Topics in Geometric Semigroup Theory and NBSAN XXXII, University of York.
25-03-2022 NBSAN XXXI: Young NBSAN, University of Manchester (Organiser).