### current teaching

The following materials relate to my current teaching.

#### $\sim$ Academic year 2018-2019, UCLan

• Introduction to Algebra and Linear Algebra, UCLan MA1811
• first-year module, semester 2
• shared with Dr Bewsher
• lectures: 9am-11am Wednesdys, FBLT2 (Foster)
• tutorials: 11am-12pm Fridays, LE111 (Leighton)
• Further Real Analysis, UCLan MA2821
• second-year module
• lectures: 2pm-4pm Fridays, LE111 (Leighton)

#### $\approx$ Research supervision 2018, UCLan

• UURIP, Summer 2018
• title: Fraïssé and Hrushovski constructions
• UCLan Undergraduate Research Internship Programme

### farrago

#### $\odot$ Correction codes

Some abbreviations or shorthand I sometimes use when marking and providing feedback.
• qed: you're finished here, don't write any more
• bod: I'm giving you the benefit of the doubt
• ecf: this is wrong, but only because of an earlier error

### old teaching

#### $\vdash$ Academic year 2017-2018, UCLan

• Introduction to Algebra and Linear Algebra, UCLan MA1811
• first-year module, semester 1
• shared with Dr Bewsher
• lectures: 9am-11am Wednesdys, FBLT2 (Foster)
• tutorials: 11am-12pm Fridays, LE111 (Leighton)
• Cryptology, UCLan MA2812
• second-year module, semester 1
• shared with Dr Daws
• lectures: 9am-11am Fridays, LE003 (Leighton)
• Further Real Analysis, UCLan MA2821
• second-year module
• lectures: 2pm-4pm Fridays, LE111 (Leighton)
• Graph Theory, UCLan MA4823
• Special Mathematics Topics, UCLan MA4999

#### $\vdash$ Academic year 2016-2017, UCLan

• Introduction to Algebra and Linear Algebra, UCLan MA1811
• Further Real Analysis, UCLan MA2821
• second-year module
• Mathematics Project, UCLan MA3999
• Graph Theory, UCLan MA4823
• Special Mathematics Topics, UCLan MA4999
• title: Fields and Arithmetic
• fourth-year module, part of MMath (not running this year)
• project proposal

#### $\vdash$ Academic year 2015-2016, UCLan

• Introductory Statistics, UCLan MA1861
• Lagrangian and Hamiltonian Mechanics, UCLan MA2841
• URIS 2016
• title: Fraïssé and Hrushovski constructions
• UCLan's undergraduate research internship scheme
• project description
The following materials relate to my time in Leeds.

#### $\models$ Project in Mathematics, Leeds MATH3001

During my time in Leeds, I proposed the following undergraduate projects, jointly with Dr Richard Elwes.
• Diophantine Equations and Hilbert's Tenth Problem (H10).
The tenth of David Hilbert's problems from 1900 asks for an algorithm which takes as input $f\in\mathbb{Z}[x_{1},...,x_{n}]$, for any $n\in\mathbb{N}$, and outputs whether or not $f$ has a solution in $\mathbb{Z}$. Eventually it was proved by Davis-Matiyasevich-Putnam-Robinson that there is no such algorithm. In modern language: the existential theory of the ring $\mathbb{Z}$ is undecidable. This is a deep theorem and the proof is difficult! However, the main ideas of the proof are accessible. Furthermore, one can vary the problem by asking instead for solutions in a given ring $R$. Some of these generalisations are easy, and some are hard. In this project we explore the original proof (for $R=\mathbb{Z}$) and study generalisations of H10 to $R=\mathbb{C},\mathbb{R},\mathbb{Q}$ and other rings and fields. This is a very active area of research with many open problems. For example, the case of $R=\mathbb{Q}$ is a very serious unknown problem. Further reading:
• TBA
Prerequisites: no special prerequisites, only the core Pure Maths courses.

• Modal Logic, Provability Logic, and the Island of Knights and Knaves.
In 1931 Kurt Gödel published his Incompleteness Theorems, possible the most exciting theorems in Logic! Roughly, Gödel's Second Incompleteness Theorem shows that a sufficiently strong consistent theory of arithmetic cannot prove a sentence expressing its own consistency. In its usual formulation, this result is deep and technical; but alternatively it can be approached via Provability Logic, which is a fairly straightforward expansion of Propositional Logic. Propositional Logic is familiar to undergraduates that have taken Mathematical Logic I. Other themes of this project are Modal Logic (another logic based in the same language as Provability Logic) and puzzles about the `Island of Knights and Knaves'. These puzzles provide a fun but rigorous and sophisticated way of exploring Logic.