Lately, I’ve been thinking of writing a blog series on number theory and physics. I’ve been thinking a lot about this topic of late, or at least about a very small corner of what is an incredibly broad area of research. In fact, the whole business of working to find a canonical class of regulators has been my only focus of research since the autumn. (We’ve been making some great progress, and I am hopeful a series of papers is to follow).
I have a general idea of how I might organise such a blog series, with the basic parameters being renormalisation and regularisation in quantum field theory from which the focus can then be expanded. So, a reasonable entry would be analysis and summability theory, likely with an introduction to infinite series and then things like power sums, etc. This would help lay some groundwork, with Hardy serving as a nice guide. In a review of many well-known summation methods, I would also like room to discuss many interesting (subtle) peculiarities and observations from my notes – for example, like those with the Euler-Maclaurin formula or the genius of Euler in his early explorations of a bridge between the continuous and the discrete. An extended focus could also be on the concept of infinities, both mathematical and physical; it would also be nice to write about history and technical perspectives in this regard. And obviously we would need to cover some analytic number theory, with modular forms being important; there is so much here from Fourier analysis to zeta-functions and L-functions, and then also active research areas like trans-series, resurgence, and effective summation methods, not to mention some brilliant recent work as it relates to non-perturbative theory. One can therefore see that, while quantum field theory and string theory serve as the primary context from a physical perspective (and, really, as the motivation, hence this post being published on the physics side of my blog), on the mathematics side there is so much rich ground to explore. I suppose toward the peak of the mountain would be a discussion on generalised classes of regulators and, of course, as part of my ongoing love affair, ideas pertaining to some canonical class of stringy regulators, which touches some of the edges of my current studies (as it is developing). On this point, it would be nice to write about why there is good reason to hope for a class of regulators that might arise naturally from a fundamental theory, even though we may still be ignorant of the full picture of that theory.
This is vaguely what I am thinking, with many gaps left to be filled. I think it could be a fun project, and if my PhD continues trend in the direction that it is, such a series would be helpful as a medium to continue thinking through what is a vast landscape of concepts. There is so much scope and so many things that could be covered, with the possibility of many distractions. The biggest difficulty would be in maintaining some modicrum of focus. What is clear is that, however it comes into form, it will take time to put together.
I’ll keep thinking about it.