Rigid structure in noncommutative, geometric and combinatorial problems

The proposed research of this proposal is in representation theory, a field of pure mathematics with strong interactions with other sciences including computing science, chemistry and physics. A basic idea of mathematics is to distill the most crucial properties from naturally occurring phenomena, leaving simply the essence of the situation to be studied and understood. We all know this idea already: long ago people did not think of numbers as abstract quantities, but as way of describing specific quantities. One sheep, two sheep, three sheep. It was a remarkable step forward to think of numbers abstractly: they are objects to which we can do things - we can apply algebraic operations to them, such as adding and subtracting, we can compare them, etc - but they are not always objects that we can visualise or specify in the real world. What does the number 1 billion look like? 1 trillion? We know that they are big numbers because we can compare them to other numbers, but how big? Our feeling for such numbers comes essentially from our ability to treat them just like any other number, and in particular ones for which we do have a very good intuition. Abstraction abounds in mathematics. For instance, the study of symmetry is encoded abstractly in the notion of a group. Groups are collections of elements which satisfy certain axioms, axioms which obviously hold for symmetries. However, a group is abstract by definition and need not be presented as symmetries of any particular object, but only as an object satisfying the given list of axioms. The axiomatic approach is a very powerful method which, in this instance, allows one to prove many general theorems, all of which can be applied to any group. Given the general abstract definition of a group, one is lead to think about simple groups, the building blocks from which every group can be built. For a long time group theorists wanted to classify simple groups, and around twenty five they came up with a comprehensive list. Many items on the list had been known since the birth of group theory, but there were a small number of exceptions. These were new groups, absolutely fundamental since they were basic building blocks, but they had not been observed earlier as symmetries of some well-known mathematical object. Where did they come from? Were they symmetries of something? This is where representation theory comes in: it studies how a group (or other abstract mathematical structures) can be the symmetry of some naturally occurring object. It weds mathematical reality and abstraction.Representation theory is thus a powerful tool that is of interest to researchers in many different fields. Within pure mathematics it is important when studying abstract systems, but it is also a very useful for understanding the orginal objects which display the symmetry. It is also used further afield, for instance in chemistry to help to study the symmetry of molecules, in physics when studying the nature of space, or in fluids to help to solve differential equations. In this proposal I intend to build mathematical tools using the rigid structure arising from the representation theory of noncommutative algebras which can then be applied to solve problems in a number of different fields and which will also be of intrinsic interest to representation theorists. In doing this, I will interact with researchers from many different topics, both in the UK and abroad. This activity will have benefits for mathematics in Edinburgh, the UK and beyond.