Mathematics at Leeds University

Paul Martin's Algebra Projects Links Pages

UNFINISHED PAGE!!...WORK IN PROGRESS... tangle category project (Summer)

A diagram category is a rather beautiful kind of generalisation of the idea of a finite group, such as the symmetric group. Like groups, these algebras arise in Physics. But while groups describe the symmetries in Physical systems, diagram categories describe the (mathematical) model of the systems themselves.
An overarching example of a diagram category is the partition category described here . Just like the symmetric groups have many subgroups, so the partition category has many subcategories. If one puts aside the Physical applications for a moment, then it becomes a challenging exercise to try to find new subcategories with interesting structures. Many such substructures have been found. But there are certainly many more still to be discovered!
A tangle category is a similar kind of idea... but with geometric topology thrown in!
Project themes:
Keywords: Doing algebra with pictures
Links: (meanwhile, if you are interested, just ask me!).
For some clues about the connections to Physics see Chapter 11 here Potts models and related problems in statistical mechanics.