Research Statement

Phenomenology

Formal theory,

Pure mathematics

Types of questions 

Let M=an area of mathematics or formal theory.

Let P=an area/aspect of HEP

Then there are 3 types of questions I look at (in descending order of how enthusiastic I am about them):

How can P be proven using M?

How can P be understood using M? 

 How can we model P utilising  M?

For example: How can all anomaly-free u(1) extensions be found using number theory.

For example: How can Inverse Higgs constraints be understood using category theory.

For example: How can we construct a model hierarchy problem using group theory?

Mathematical skills

By the standards of: I would like to think I could use these (broad) areas in my research, and spot where they would be useful to use.

Category Theory

Lie groups

&

Lie algebras

Number
theory

Differential geometry

Ring theory

Physics skills

Gauge 

groups

GUTs

&

Symmetry

breaking

Inverse Higgs

constraints

Anomaly
cancelation

Any conversation with a high energy physicist no doubt contains the question of "what sorts of things do you work on?". My answer to this question is somewhat chaotic, for instance although I model build, but I wouldn't count myself as a model builder. In truth, I work any project where my skill sets can be put to good use. So what are my skills? Over the years, I have taken upon myself to learn areas of pure mathematics which can be applied in a useful way to HEP, and which are not part of the usual physicists curriculum.  For instance, differential geometry, number theory, the theory of rings and category theory. You may be asking yourself, "sure but is category theory really useful in physics"? I reassure you, that the same questions were being asked about group theory before its acceptance by the community.

Here I will not detail the specifics of projects I plan to work on in the future. Not because I worry about putting them online, but because I don't want to have to rewrite this  page every time I come up with new ideas. So let me give the basic flavour of my current and planned work.

Phenomenology

My work in pheno has primarily been focused on anomaly-free gauge theories and gauge extensions of the Standard Model. These sorts of projects tend to focus on the classification of certain subsets of gauge algebra extensions. Unsurprisingly, the theory of Lie algebras is at the forefront of this work, as well as (perhaps more surprisingly) number theory. 

 

occasionally do more traditional model building. This is either related to previous work on gauge algebras, or completely tangental to it. On these sorts of projects my main contribution tends to be more oriented towards the correct mathematical formulation of the model, rather then the actual comparison to experiments (which I leave to other more qualified people) - i.e. I rarely contribute to the `pheno' section of papers. That said, I have been known to do the odd comparison to experiments. 

More formal 

In my work, I also focus on more formal aspects of HEP. Many concepts in phenomenology are known only at an approximate level. This is, due in part, due to the lack of a formal definition of the path integral. Certain quantum theories can be understood formally in the frame work of category theory, these are refereed to as topological quantum field theories.  By understanding what happens for these simpler theories, one can make comparison with more sophisticated QFTs. This not only aids in understanding existing concepts in pheno, but also allows you to ask the question of - what else can be done?