[This post has a technical-sounding title but is not technical in its contents. That will come in follow-up posts; I generally think that mathematicians should take the reader to dinner, first;-).]
When I studied mathematics in my master’s, I conformed my interests, for practical purposes, to those of my supervisor. This did save some anguish in choosing the topic for my thesis, but it meant that the only soul I had in the game was that of the artisan who wants to do things as well as possible. Maybe I was just too sensitive, but pushing so hard in an area that didn’t at least start out by sparking my curiosity was subtly damaging; I lost my appetite for mathematics, to a degree. It’s important to realize that no project exhilarates at all points in time, but I think there needs to be at least be a moment or two of that amid all the problem-crunching to make the use of life worth it.
Once I joined the space sector, there was a flood of topics that piqued my curiosity. Many were found in the course of my work, but others were found in becoming more of an adult and interested in the goings-on of the world. I’ve spent the last couple of years exploring all of that, and I continue to do so. At the same time, my mathematical curiosity has “healed” enough that I’ve circled back to one of my long-time interests: scaling.
Happily, a lot of laypeople can at least recognize a fractal at-sight. In fact, you’ve probably seen a T-shirt like this one before:
Fractals are self-similar structures across different scales. For some reason, they have captured the popular imagination enough to make it into the Frozen song, “Let It Go”, among many other contexts. Fractals do have the excellent quality of being a very useful abstraction for understanding objects that we encounter in daily life, such as waves and trees. On the other hand, I don’t personally find them all that interesting, elegant as they are. One way of thinking about fractals is that they have a special type of invariance, or symmetry: Their structure remains the same as the scale at which they exist changes. Now, I find it very intriguing what conditions enable fractal patterns to exist in nature. But as far as structures, themselves, go, I find it far more intriguing when the structure of a system is partly or completely changed as the scale at which it exists changes.
My interest in this is both purely intellectual and practical. To take the practical angle, consider the Earth’s atmosphere. This very complicated and vital system occupies a range of temporal and spatial scales, all of which are of central importance to humanity. You probably care about what the troposphere, the atmospheric layer responsible for weather, is doing over your commute to work today. You might also care about whether the atmosphere, as a whole, will be able to protect your grandchildren who will live on some other continent in a few decades. Even if you don’t, lots of other people you might have to interact with do get worked up about that.
When people try to describe a physical system using formalisms like mathematical expressions, they often do so with a particular purpose in mind, and often that purpose has to do with predicting how the system behaves under certain conditions. Thinking of the atmosphere example, we might want to predict what the local weather will be in a few days. But maybe, for the sake of our grandchildren, we want to predict reflectance by the atmosphere of solar radiation in a few decades. If the atmosphere were self-similar at all temporal and spatial scales, then we could apply the same mathematical model of the atmosphere to both questions and get satisfactory results. But this isn’t the case!
From here, my interest is captured by 1) why this is, 2) what the structure(s) is/are that operate(s) at a given spatial or temporal scale for a given system, and 3) what is required, mathematically, to get us from one “level” of the mathematics describing a system to another (If the “levels” of a physical system have to interact to produce the phenomena we see in the world—like cloud formations leading to storms—then don’t the mathematics describing them have to, also?). It’s this last point where renormalization-group theory comes especially to bear.
In the next N-number of posts, I’ll delve more deeply into this subject, probably using the Earth’s atmosphere as a touchstone example for the discussion—though I hope to also go into a few of my other pet areas, such as gravity and neuroscience and cognition.