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Papers and Publications

  1. Small Mahler Measures Coming from Digraphs - Experimental Mathematics (2021); link

Talks and Presentations

Polynomials with the same Mahler Measure
YRANT III, Virtual (hosted by University of Bristol), August 2021
 
Abstract: The Mahler measure is used to show how far away a polynomial’s roots are from being in the unit circle. We explore under what conditions two integer polynomials can have the same Mahler measure. From here, we look at a conjecture which argues that Mahler measure values are, up to a set of conditions, unique to certain polynomials. We end with a brief look at my attempts to prove this conjecture, and what the future holds.
 
Polynomials with the same Mahler Measure
YRM21, Virtual (hosted by University of Bristol), June 2021
 
Abstract: The Mahler measure is used to show how far away a polynomial’s roots are from being in the unit circle. We explore under what conditions two integer polynomials can have the same Mahler measure. From here, we look at a conjecture which argues that Mahler measure values are, up to a set of conditions, unique to certain polynomials. We end with a brief look at my attempts to prove this conjecture, and what the future holds.

Slides
Polynomials with the same Mahler Measure
BECMC21, Virtual (hosted by University of Birmingham), July 2021
 
Abstract: The Mahler measure is used to show how far away a polynomial’s roots are from being in the unit circle. We explore under what conditions two integer polynomials can have the same Mahler measure. From here, we look at a conjecture which argues that Mahler measure values are, up to a set of conditions, unique to certain polynomials. We end with a brief look at my attempts to prove this conjecture, and what the future holds.

 
Mahler Measure and Polynomial Equivalence
WISDOM Research Showcase, Virtual (hosted by Royal Holloway), November 2020
 
Abstract: In this short talk, I'll explore scenarios when two single variable integer polynomials can share the same Mahler measure. 
Limit Points of Mahler Measures coming from Digraphs
BECMC20, Virtual (hosted by University of Birmingham), July 2020
 
Abstract: The search to find the smallest Mahler measure, or a complete set of small Mahler measures, has been ongoing for almost 100 years. I will introduce a recent concept - the Mahler measure of a directed graph (digraph) - and how some tools from graph theory have helped us find limit points of Mahler measures.
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Video link (scroll down half way; Session 6, Stream 2)
Pi
Exploring Maths 2019, Royal Holloway, June 2019
This talk was for Year 12 A-Level students
Abstract: The number Pi keeps cropping up in mathematics, even if we don’t see any circles. For instance, the infinite alternating sum of the reciprocals of all odd positive integers yields Pi/4; the probability that two positive integers are relatively prime is 6/PI^2. We’ll explore a few interesting facts about Pi and how little we actually know about one of the most fundamental constants in mathematics!
Lehmer's Problem and Graphs
Postgraduate Seminar, Royal Holloway, March 2019
 
Abstract: Lehmer's (Mahler Measure) Conjecture was posed in the 1930s, and remains unsolved, despite many number theoretic attempts to do so.
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I introduce a link between Lehmer's Problem and certain types of graphs, which hasn't been explored in much detail yet. This leads onto work I am currently doing, and some results which I have currently obtained.
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Slides
Non-Monogenous Orders and Essential Discriminant Divisors
Tomorrow's Mathematicians Today, University of York February 2015
 
Abstract: 
Assuming just the definition of a ring, in the first part of this talk, I will give a relaxed and basic introduction to algebraic number theory; a major branch of number theory. This will include lots of interesting examples; some of which might be surprising!
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The second part of the talk will focus on non-monogenous phenomena, which is when an algebraic structure (say a ring) is not generated by one element. This will bring us nicely to some results that I found whilst doing a small research project in Summer 2014.
Slides
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Limit Points of Mahler Measures coming from Digraphs
Pure Maths Seminar, Royal Holloway, March 2020
 
Abstract: The Mahler measure of a single variable polynomial f can be defined as the geometric mean of |f(z)| on the unit circle. The search to find the smallest Mahler measure, or a complete set of `small' Mahler measures, has been ongoing for almost 100 years. I will introduce a recent concept - the Mahler measure of a directed graph (digraph) - and how some tools from graph theory have helped us find limit points of Mahler measures.

Slides
Mahler Measure of Graphs
PhD Mini Conference, Royal Holloway, April 2019
 
Abstract: I explore the connection between Mahler Measure and graphs. I'll discuss some recent results I've found and show some families of graphs which have small Mahler measures.
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Slides
Analytic Number Theory and the Goldbach Conjecture
Warwick Maths Society, February 2016
 
Abstract: I'll give a brief overview of some of they key introductory results and theorems in Analytic Number Theory.
 
I'll then go into detail about the Goldbach Conjecture - a problem that sounds even more simple than Fermat's Last Theorem, but still remains unsolved!
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An Introduction to Algebraic Number Theory
Warwick Maths Society, January 2015
 
Abstract: Algebraic Number Theory is a fascinating subject, but has a reputation for requiring a lot of technical algebraic and number theoretic background to understand.
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Though strictly speaking this is true, in this talk I'll give an overview to some of the most interesting results from the area, using little more than basic 1st year algebra results.
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An Introduction to p-adic Numbers
Warwick Maths Society, November 2014
 
Abstract: 
In this talk, I aim to give a brief introduction to what p-adic numbers are, and why they are important.  There will be plenty of examples, all of which will help illustrate the equivalence of the various definitions and constructions of p-adic numbers.
 
The end of the talk will be about the usefulness of the p-adics.  In particular, about how they crop up outside Number Theory.  We will look at how they are used to prove Monsky's Theorem - a combinatorial problem.
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Places

​The following is a (not-complete) list of Conferences, Summer Schools, etc that I have attended, as well as future places where I'll be.
Ones in italic are where I have spoken / presented:
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  • ERLASS V, Virtually (hosted by RHUL and UCL), December 2021 
  • ERLASS IV, Virtually (hosted by UCL), September 2021 
  • YRANT III, Virtually (hosted by University of Bristol), August 2021
  • BECMC21, Virtually (hosted by University of Birmingham), July 2021
  • YRM21, Virtually (hosted by University of Bristol), June 2021
  • ERLASS II, Virtually (hosted by UCL), December 2020
  • WISDOM Research Showcase, Virtually (hosted by Royal Holloway), November 2020
  • British Early Career Mathematician Colloquium , Virtually (hosted by University of Birmingham), July 2020
  • CANTA Launch, Virtually (hosted by Royal Holloway), June 2019
  • ERLASS I, Royal Holloway, December 2019
  • Cryptography Meets Graph Theory, Universität Würzburg, July 2019
  • Exploring Maths 2019, Royal Holloway, June 2019
  • PhD Mini Conference, Royal Holloway, April 2019
  • WIMP Spring Meeting, Imperial College London, March 2015
  • Tomorrow's Mathematicians Today 2015, University of York, February 2015
  • Tomorrow's Mathematicians Today 2014, University of Surrey, February 2014
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