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Mathematics at Leeds University
Paul Martin's Algebra Projects Links Page
Diagram algebra/category project
A diagram category is a rather beautiful kind of generalisation of
the idea of a finite group, which arises in Physics.
Project themes:

What are they?

How could they be explained to a wide audience of clever nonalgebraists?

How can their "pictorial" nature be used in rigorous calculations?
 How do they connect with their applications?
 ...
Keywords: Doing algebra with pictures
Links:
(meanwhile, if you are interested, just ask me!).
KazhdanLusztig polynomials
The polynomials that we have in mind have a beautiful
(and in principle relatively simple) iterative construction
which combines many aspects of algebra and Euclidean geometry.
They have applications in algebraic representation theory.
Project themes:

What are they?

How could they be explained to a wide audience of clever nonalgebraists?

How can their "geometrical" nature be used to help visualise them?
 How do they connect with their applications? (For example
in the study of "tilting modules" in representation theory.)
 ...
Keywords: Fun with polynomials, combinatorics and geometry
Links:
(meanwhile, if you are interested, just ask me!).
Polynomials of Potts
The Potts model is a beautifully simple model of a complex manybody
system (such as a bar magnet, or the entire universe).
In statistical mechanical modelling the fundamental mathematical
object associated to a given model
is the partition function,
and for the Potts models the partition function is
a polynomial.
This polynomial is usually so huge that there is nothing you can do
with it. But sometimes we are very lucky and we can compute some of
these polynomials.
The next question that arises is:
what can we learn from them? How can we analyse them?
For a quick "answer"
 or rather an intriguing glimpse 
see Chapter 11 here
Potts models and related problems in statistical mechanics.
Keywords:
Fun with polynomials as mathematical models in Physics
Links:
(meanwhile, if you are interested, just ask me!).