# Successful grant: on the Product Decomposition Conjecture

I have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “first grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “The Product Decomposition Conjecture”.

The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.

Excitingly, the university has agreed to fund a PhD student as part of this research. I just drafted a short description of what the PhD would be about, and I’ll post this below. (Note that this description might be edited a little over the next few days. In any case, it should give an idea of what the project will be about.) **If you are interested, please get in touch!**

This programme of research is within the study of finite group theory (although some investigation of linear algebraic groups may also be involved). The aim is to prove, or partially prove, the Product Decomposition Conjecture which concerns “conjugate-growth” of subsets of a finite simple group: roughly speaking, given a finite nonabelian simple group G and a subset A in G of size at least 2, we would like to show that one can always write G as a product of “not many” conjugates of A.

This notion of conjugate-growth has interesting connections to many interesting areas of mathematics, including expander graphs, the product growth results of Helfgott et al, bases of permutation groups, word problems and more.

In the process of working on this conjecture, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.