Here at the WMS we are always thinking about how to help with your studies, and this is our latest venture. The following collection of second year essays are to complement the (more dated) examples given on Mathstuff. The essays below are just a few examples of what essays look like... Some will scare you with their length (whilst others will leave you giggling at how short they are), and whilst you may be baffled by the complexity of some topics, you will realise that this module can be a fun and interesting way to discover new maths.

Most of the advice that can be given about the essay is self evident, but is worth repeating.

Choose a topic that you are interested in: if you are engaged by your subject the reading and writing won't be nearly so much of a chore.

Writing the essay at the correct level is something that you have to guess at a bit: if you can understand everything you read on first go through, then perhaps try and push yourself a bit harder. If you are blindly copying from a book because you cannot understand the proof, then either seek help, or seek a new idea.

Present your essay clearly, and professionally. A good bibliography always makes an essay appear well penned, whilst good spelling and structure will make the markers much happier to keep reading. Do not be fooled however, a well laid out essay about school-level long division will not get you good marks (if this is your case, reread the above comments about understanding everything on first go...)!

So, enjoy the following collection. You can use it to see what sort of things write about, how difficult/accessible the essays can be, how long they are, or just to read at leisure to learn about new maths: after all your essay should be accessible to people at your level too.

Second year essays:
Homology (First, 92%)
The Axiom Of Choice (First, 89%)
The Gamma Function (First, 88%)
Algebraic Curves, Projective Mappings and Bezout’s Theorem (First, 85%)
Polynomials and Their Application to Ruler and Compass Constructions (First, 85%)
Sperner's Lemma and Other Topics in Combinatorial Topology (First, 84%)
Single Server Queueing Systems (First, 80%)
Set Theory and the Axiom of Choice (First, 79%)
On Sums of Integers (First, 79%)
Transcendental Numbers (First, 78%)
Sampling in Harmonic Analysis (First, 78%)
What is a Cardinal Number? (First, 78%)
Gröbner Bases for Polynomial Ideals (First, 75%)
The Logistic Map, Period Doubling Cascades and the Onset of Chaos (First, 75%)
Connections between Mathematics and Chemical Kinetics - Experimental and Theoretical (First, 70%)
An Introduction to the Gamma Function (2:1, 69%)
Hilbert Spaces (2:1, 69%)
An Introduction to Complex Analysis Through Dynamical Systems (2:1, 69%)
The History and Applications of Pell's Equation (2:1, 68%)
How to choose? - An Introduction to Decision Theory (2:1, 68%)
The Weil Conjectures (2:1, 66%)
Quaternions And Their Importance (2:1, 65%)
Diophantine Equations: Integer Solutions (2:1, 62%)

Also see the mathstuff page which has more examples.

Excursions essays:
Curvature (First, 100%)
Generalised Abstract Nonsense (A Short Introduction to the Theory of Categories) (First, ~89%)
Orthonormal Bases in Hilbert Spaces (First, 88%)
On the Sequence of Prime Numbers (First, ~80%)
The Gamma Function (First, ~79%)
The Inverted Pendulum (2:1, ~70%)
An Introduction to p-adic Analysis (2:1, ~65%)