- don't forget to visit the
**smörgåsbord**of fun and recreational maths!

- or visit me during my office hours

- the course handbook for BSc and MMath (academic year 2015-2016)
- BSc course page
- MMath course page
- blackboard

- 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)

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

- title:

- The Rank-Nullity Theorem is a note written for a student at UCLan in 2017.
- Arithmetisation of syntax is a note from a course on Gödel's Incompleteness Theorems that I gave in the Nesin Mathematics Village in 2013.
- The Cayley-Hamilton, some thoughts about the proof is a note that came from a discussion in tutorials at Merton.

- Riemann Surfaces (extended text), a 10-minute lecture.
- Riemann Surfaces, problem sheet 1.
- Here is a draft course outline for a graph theory course that I prepared for an interview in 2015.
- 10-minute lecture on Cauchy Sequences is the extended text of a lecture I prepared for an interview in 2015. Here is a slightly earlier version.
- 10-minute lecture on Series is the text of a lecture I prepared for an interview in 2015.

: you're finished here, don't write any more*qed*: please read the question*rtq*: I'm giving you the benefit of the doubt*bod*: this is wrong, but only because of an earlier error*ecf*

- 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
- fourth-year module, part of MMath
- lectures: 11am-1pm Thursdays, LE111 (Leighton)
- original course design

- Special Mathematics Topics, UCLan MA4999
- title:
**Further theories of integration on the real line** - fourth-year module, part of MMath
- project proposal
- project specification

- title:

- Introduction to Algebra and Linear Algebra, UCLan MA1811
- first-year module
- course plan

- Further Real Analysis, UCLan MA2821
- second-year module

- Mathematics Project, UCLan MA3999
- title:
**Gödel and Turing - Incompleteness and Uncomputability** - third-year module
- project proposal
- project specification (for the Gödel version)

- title:
- Graph Theory, UCLan MA4823
- fourth-year module, part of MMath
- the original course design.

- Special Mathematics Topics, UCLan MA4999
- title:
**Fields and Arithmetic** - fourth-year module, part of MMath (not running this year)
- project proposal

- title:

- Introductory Statistics, UCLan MA1861
- first-year module
- here are some
**skeletal lecture notes**.

- Lagrangian and Hamiltonian Mechanics, UCLan MA2841
- second-year module, semester 2
- here are some
**skeletal lecture notes**.

- URIS 2016
- title:
**Fraïssé and Hrushovski constructions** - UCLan's
*undergraduate research internship scheme* - project description

- title:

**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.- Here is a
**short introduction**to the topic*When can we solve Diophantine equations?*

*TBA*

- Here is a
**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.

Further reading:*Forever Undecided: A Puzzle Guide to Gödel*, Raymond Smullyan.*Logic*, Wilfrid Hodges.