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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:
Keywords: Doing algebra with pictures
Links: (meanwhile, if you are interested, just ask me!).

Kazhdan-Lusztig 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:
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 many-body 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!).