Research

I completed a PhD at the University of Leeds, and my supervisor was John Truss. During my PhD, I studied reducts of countable homogeneous structures.

What is a reduct? Loosely speaking, if you have two structures A and B with the same domain (but potentially in different languages), then A is a reduct of B if A is a less detailed version of B, or, if A is obtained by discarding information from B.

What is a homogeneous structure? Very roughly, it says that the structure looks the same no matter where you are located inside the structure. A couple of (non-mathematical) friends of mine insisted that I try to explain what my research was about, so I made an attempt at describing what homogeneous structures are, with some help from the cast of West Side Story. The result is this article.

What am I doing with reducts of homogeneous structures? Specifically, I am trying to determine all the reducts of some particular structures. I have been able to determine the reducts of the countable vector space over F_2, using the classification of countably-categorical strictly-minimal structures. (This strategy was suggested to me by Dugald Macpherson.)

In 2014 I completed the task of determining the reducts of the generic digraph, which can be defined as the unique, up to isomorphism, countable homogeneous digraph that embeds all finite digraphs. The proof uses the classification of the randam graph (by Simon Thomas) and of the random tournament (by one of his students Bennett) and the idea of `canonical functions' (developed by Manuel Bodirsky and Michael Pinsker).

In 2015, joint with Michael Kompatscher, the task of determining the reducts of the Henson digraphs was completed. Henson digraphs are countable homogeneous digraphs which forbid some non-empty set of finite tournaments. This work has been submitted to the Journal of Symbolic Logic and is being refereed. The strategy for this is the same as for the generic digraph. A corollary of this work is an example of uncountably many maximal non-conjugate closed subgroups of Sym(\mathbb{N}).

Published items

  • Contiuum maximal-closed subgroups of Sym(N) via the classification of the reducts of the Henson digraph
    Submitted
    arxiv.org/abs/1509.07674.

Miscellaneous items

  • 2013 May, Stable Groups. (Joint with Ricardo Bello Aguirre and Daniel Wolf)
    Lecture notes from a graduate course given by Anand Pillay at the University of Leeds.
  • 2013 January, Homogeneous Structures.
    This was my attempt at satiating the curiosity of a couple of non-mathematical friends.
  • 2012 February, What is a model of set theory?
    I wrote this for a friend who was curious about the circularity of sets being used as models of set theory.

Conferences and Events

  • CETL-MSOR Conference, University of Greenwich, 2015 September. ("Continuing Excellence in the Teaching and Learning of Mathematics, Statistics and OR")
  • Leeds Logic Day, University of Leeds, 2015 March. An event I organised. It combined the inaugural Leeds PhD Logic Day and the Cambridge-Leeds Logic Seminar.

Talks

  1. Non-forking spectra, Model Theory Month in Muenster Conference, May 2016.
    Given a first-order theory T, the stability function f is a function which maps a cardinal \kappa to the cardinality of the maximum number of types over a model of T of size \kappa. For example, if T is the theory of DLO, then f(\aleph_0) = 2^{\aleph_0}. The stability function is well-understood. The non-forking spectrum of a theory T is a generalisation of the stability function which takes as inputs two cardinals \kappa and \lambda, and outputs the maximum number of 1-types over a model of size of \lambda that do not fork over a submodel of size \kappa. In this talk, I will present results obtained by Chernikov, Kaplan and Shelah about non-forking spectra. (The results were first put on Arxiv in 2012, but a second version was uploaded in Aug 2015).

  2. Morley's Theorem, Leeds Model Theory Set Theory PG Seminar, Nov 2015.
    Let L be a countable language and let T be a complete L-theory. Morley's Theorem states that if T is \kappa-categorical for some uncountable \kappa, then T is \kappa-categorical for all uncountable \kappa. In this talk I will present a sketch of a proof of this theorem, with the aim that you come away with some of the main ideas and steps involved.

  3. Morley Rank, Leeds Model Theory Set Theory PG Seminar, Oct 2015.
    I will try to introduce the notion of Morley rank, perhaps leading to how it links to stability.

  4. Infinitely many primes, Leeds Pure PG Seminar, Oct 2015.
    In this talk, I will aim to present proofs of the following.
    Theorem 1. There are infinitely many primes.
    Theorem 2. There are infinitely many primes.
    Theorem 3. There are infinitely many primes.
    Theorem 4. There are infinitely many primes.
    Theorem 5. There are infinitely many primes.
    Theorem 6. There are infinitely many primes.

    Time permitting, I will also present.
    Theorem 7. There are infinitely many primes.
    Joking aside, I aim to present seven different proofs that there are infinitely many primes.

  5. Uncountably many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs, Higher Infinite Programme at the Isaac Newton Institute, Oct 2015. (This talk was recorded and the video is available in the link).
    This work contributes to the two closely related areas of countable homogeneous structures and infinite permutation groups. In the permutation group side, we answered a question of Macpherson that asked to show that there are uncountably many pairwise non-conjugate maximal-closed subgroups of Sym(\mathbb{N}). This was achieved by taking the automorphism groups of uncountably many pairwise non-isomorphic Henson digraphs. The fact these groups are maximal-closed follows from the classification of the reducts of Henson digraphs. In itself, this classification contributes to the building list of structures whose reducts are known and also provides further evidence that Thomas' conjecture is true. In this talk, my main aim will be to describe the construction of these continuum many maximal-closed subgroups, which will include Henson's famous construction of continuum many countable homogeneous digraphs. Any remaining time will be spent giving some of the ideas behind how we prove these groups are maximal closed.

  6. Continuum many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs,Leeds Algebra, Logic and Algorithms Seminar, Sep 2015.
    This work contributes to the two strongly related areas of countable homogeneous structures and infinite permutation groups. In the permutation group side, we answered a question of Macpherson that asked to show that there are continuum many pairwise non-conjugate maximal-closed subgroups of Sym(N). This was achieved by taking the automorphism groups of continuum many Henson digraphs. The fact these groups are maximal-closed follows from the classification of the reducts of Henson digraphs. In itself, this classification contributes to the building list of structures whose reducts are known and also provides further evidence that Thomas' conjecture is true. In this talk, my main aim will be to describe the construction of these continuum many maximal-closed subgroups, which will include Henson's famous construction of continuum many countable homogeneous digraphs. Any remaining time will be spent giving some of the ideas behind how we prove these groups are maximal closed.

  7. Uncountably many maximal subgroups of Sym(w) via reducts of Henson digraphs, LMS-EPSRC Durham Symposium, Jul 2015. (This talk was recorded and the video is available in the link).
    Macpherson asked whether there are uncountably many maximal closed subgroups of Sym(w), where G is maximal means that G is not equal to Sym(w) and there are no closed subgroups in between G or Sym(w). In this talk, I will present a positive answer to this question using Henson digraphs.

  8. Godel's Incompleteness Theorems, Leeds Pure PG Seminar, Apr 2015.
    Heisenberg, Godel, Turing, Chomsky, a knight, a knave, Cameron, Miliband and Farage walk into a bar. The barman asks them if they think this joke is funny. Heisenberg is uncertain. Godel says its impossible to know since they're inside the joke. Turing couldn't decide. Chomsky says that it is of course funny, but it has just been told wrong. The knight says yes. The knave says no. Cameron says the joke has been getting funnier over the past 5 years. Miliband says Cameron is wrong. Farage says it would be funnier without Heisenberg, Godel and Chomsky. Pauli was upset because of his exclusion from the joke. By the end of the talk, you should understand at least one tenth of this joke.

  9. Explicit definition of structures from linear orderings, Leeds PG Model Theory Seminar, Apr 2015.
    I will discuss the notion of explicit definitions in linear orderings, from Lachlan's paper Structures coordinatised by indiscernible sets

  10. Model Companions, Leeds PG Model Theory Seminar, Feb 2015.

  11. The Reducts of the Generic Digraph, North British Semigroups and Applications Network, York Jan 2015.

  12. The 11 Reducts of the Generic Digraph, Lancashire-Yorkshire Model Theory Seminar, Jan 15.
    Given two structures M and N, we say that N is a reduct of M if, intuitively speaking, N is a less detailed version of M or if N is obtained from M by discarding information. In this talk, I will describe what the reducts of the generic digraph are and time permitting will describe some aspects of the proof.

  13. The Reducts of the Generic Digraph, British Postgraduate Model Theory Conference, Oxford Jan 2015.
    Loosely speaking, a structure N is a reduct of a structure M if N is a less detailed version of M, or, if N is obtained by discarding information from M. The usual set-up is that a structure M is given and one wants to describe the reducts of M. In this talk, I will present work done on determining the reducts of the generic digraph.

  14. Junker and Zieger's proof for the reducts of the rationals, Leeds PG Model Theory Seminar, Oct 2014.
    I will present Junker and Ziegler's proof that (Q,<) has 5 reducts.

  15. Homogeneous Structures, Cambridge Logic Event, Oct 2014.
    The study of homogeneous structures provides a meeting point for model theory, combinatorics and permutation group theory. There are also connections with topological dynamics via structural Ramsey theory, and with constraint satisfaction problems in complexity theory, via universal algebra. My own research is in relation to Simon Thomas' conjecture, which states that any (countable) homogeneous structure in a finite relational language has only finitely many reducts.\\\ In this talk, I aim to give you a flavour for what homogeneous structures are, with an emphasis on providing examples. I will end by describing a few questions that people tackle in this area.

  16. Alternative Set Theories, British Logic Colloquium PhD Day, Sep 2014.
    The majority of mathematicians know about ZF(C) Set Theory and some may even be able to list its axioms. However, what is the purpose of these axioms? What is the purpose of Set Theory? By thinking about these questions, one is naturally lead to consider if there are any alternatives. In this talk, I will informally discuss these questions and briefly describe two or three alternatives.

  17. Reducts of the Generic Digraph, Cambridge-Leeds Logic Seminar, Apr 2014.
    I will describe what reducts are, what the generic digraph is, and end by describing the reducts of the generic digraph.

  18. A few pages from Anand's book, Leeds PG Model Theory, Mar 14.
    I will go over the definition, a few examples and a few basic results of heirs and coheirs.

  19. WQOs, Leeds Pure PG Seminar, Feb 2014.
    An introduction to WQO theory will be given via its use in the solution of a question proposed by Erdos, namely, Problem 4358 (Amer. Math. Monthly, 56 (1949), 480)

  20. Structures with the same lattice of reducts, Cambridge-Leeds Logic Seminar, Nov 2013.
    I will start by describing what reducts are and then describe four structures which, mysteriously, all have the same lattice of reducts.

  21. Reducts of the Generic Ordered Graph, Leeds Model Theory Seminar, Nov 2013.

  22. Introduction to Introductory Stability Theory, Leeds PG Model Theory

  23. Finite Ramsey Theory, Leeds Pure PG Seminar, Oct 2013.
    Loosely speaking, Ramsey Theory is about determining whether you can find order in a large amount of disorder. The most widely known example is that of people in a party: if there are at least 6 people at a party, then you can find three people who have all met each other before, or, three who have not met each other before. In this talk, I will prove the Finite Ramsey Theorem, which generalises the result described, and Van der Waerden's Theorem, which concerns coloured beads on a string. I will also discuss corollaries and generalisations of these results, including a non-trivial alternative to Naughts and Crosses (or Tic-Tac-Toe).

  24. Reducts of Aleph_0 Categorical Structures, British Logic Colloquium PhD Day, Sep 2013.

  25. Reducts of Ramsey Structures, Cambridge-Leeds Logic Seminar, Jun 2013.

  26. Ramsey Theory, Leeds Pure PG Seminar, Mar 2013.
    I will give an introductory talk on Ramsey theory, in which (depending on how I manage the time) I will present some subset of {Ramsey's Theorem and its proof, `applications' of Ramsey's Theorem, Schur's Lemma, Van der Waerden's Theorem, generalisations of Ramsey's Theorem, generalisations of Van der Waerden's Theorem, a fun (and surprisingly non-trivial) alternative to Naughts & Crosses}.

  27. Introductory Stability, Leeds PG Model Theory, Dec 2012.
    I will present the definition of stable theories and definable types, and sketch how the two notions coincide.

  28. Godel's (1st) Incompleteness Theorem, Cambridge Part III Seminar Series, Dec 2011.