Jonathan Hickman

Jonathan’s pre-lecturer life

Born in Hartlepool, England, Dr. Jonathan Hickman found himself in the Scottish highlands as a high schooler. Coming out of there he went straight in to the University of Edinburgh to get a degree in MA Mathematics. He then went on to do a masters at Cambridge, before ending up back in Edinburgh for his PhD, which he completed in 2015.

“After that things got a bit more interesting,” he says; he found himself in the US where he held post-doctoral positions at the University of Chicago and the Mathematical Sciences Research Institute, Berkeley.

The US was a good place to be in for his area of research, he goes on to explain, since the modern theory of Harmonic Analysis (his field of research) started in the middle of the twentieth century with a couple of mathematicians working on it in Chicago.

After this short break from Scotland, he found himself back again when he took up another post-doctoral position at the University of St. Andrews. Finally from there he came full circle; last year when he took up a lectureship role at the University of Edinburgh.

 

Jonathan’s passion for Harmonic Analysis

Harmonic Analysis, “at the heart of it, is understanding the Fourier transform”, he says. The field has a number of applications; recently it has been used to study large data sets and matrices, though Jonathan’s research, in particular, focuses just on pure analysis.

His interest in this particular field first developed through a project that he had to do as an undergraduate. Before that he was more interested in other fields, especially differential geometry. During his time at university, he further found himself leaning towards a number of different mathematical areas such as combinatorics and dynamics, “but in the end, I came back to Harmonic Analysis.”

 

“I would say there’s at least three reasons”, he reveals, when asked about why he finds Harmonic Analysis interesting.

“First of all, I like analysis, in general, because there are many different ways to attack a problem.” He finds it more flexible compared to other fields, especially because when solving a problem one doesn’t need too much background information to get started. “You can kind of get your hands dirty in analysis,” problem solving can be a bit more rigid in fields like abstract algebra, “but I mean I don’t really like to make these distinctions too much.” He believes it’s always better to have various tools at hand from a number of different fields when doing maths.

“The second reason I like harmonic analysis is that it interacts with lots of other things.” The fact that it is not isolated from other fields and is interconnected with a number of other topics in mathematics also makes it more appealing to him.

“The third reason is that it’s extremely topical.” Some fields become stagnant after a point for various reasons and aren’t as active, but there are a lot of new ideas floating around right now in his area of research. There’s been huge breakthroughs and the understanding of many problems has advanced in the last few years, “so,” he says, “it’s really exciting to be a part of a field that’s blowing up like that.”

“There’s still some big problems that we’re really far from understanding”, he continues. There have been advances that have helped solve some longstanding problems, though the tools that are now available have mostly helped give insight into why some problems are really hard and, while not quite solved, have provided more information about them. “Basically, all the core problems in my area are kind of really, really, far from being solved.”

 

Life as a researcher

“Research is a lot less passive in some ways”, he says, when comparing studying mathematics to doing research. It seems to require more attention and is much more rigorous; the degree of the difference can be seen in the amount of time spent on a single problem and also the extent of the output or solution. “I spent two years on a research problem, and the solution was like, a hundred-page paper, so the order of magnitude is much higher.”

“Also, in a psychological sense, I think when people start to do mathematical research, they don’t have much confidence,” he goes onto explain. People are unsure of whether what they’re trying to show is true or false, and there’s uncertainty as to whether their method is going to work. When doing a question on an exercise, on the other hand, there’s always the certainty that there is a solution, and even when one might not understand the problem well, it isn’t a big deal. “But if you can’t get to grips with your research problem, and you’ve invested months doing this thing, then that is a big deal.”

Reflecting on the transition into research for him, he says, “I really like learning mathematics, so when I started my PhD, at least in the first year, I spent way too much time reading other people’s things, because I was in this mindset of a student.” After a point, he believes that that isn’t enough when doing research; independent thinking as well as the active application of the knowledge becomes more important.

 

It is also difficult, he believes, to find topics to research on as a PhD student when the chosen field isn’t as familiar. Though it is only a problem in the beginning, because over time and after interacting with a number of concepts in the field, the topics come easier. For now, the struggle he faces is to find time to get to explore all the ideas that he wants to, “I have a list of probably about 20 topics that I would like to work on, but I just don’t have the time.” He finished a paper recently, and though it had a satisfying result, more questions come up automatically; related to the limitations and applications of the result. So, there’s always something to work on. Though, he says, it isn’t ideal to randomly jump from one topic to another, “It’s good to have a central research program, at least in my experience, where things are a direct continuation of one another.”

At the same time, he thinks it’s good to branch out and spend some time doing different things sometimes, “just to broaden your experience.” So, it’s about finding the balance between the two.

“I mean if you’re Terry Tao or someone, you can work on lots of different things,” but Jonathan likes to have a general theme in his work and most of it revolves around his speciality.

 

Jonathan’s thoughts on being a lecturer

Coming to his current teaching roles, we find out he’s a lecturer for three courses. One on Harmonic Analysis for the MMath program called Real Analysis, which he says he has had to do quite a bit of work for. He’s also teaching a course for PhD students, which is based around analysis and measure theory. It’s shared across different universities around Scotland, where one lecturer out of a few does a set number of lectures for one part of the course.

And finally, he’s teaching an undergraduate course called Facets of Mathematics, which he seems particularly excited about. It has three components and he’s a lecturer for the Pure Maths one, this deals with basic Algebraic Geometry, “It’s quite an interesting course, I quite like it. I hope the students like it as much as I do.”

 

Research vs Teaching

Moving on to the hard questions, when asked about his preference between research and teaching, he says, “I like research more if I was forced to answer that but that said I would not like to do just research.” He likes teaching almost as much; it’s a learning experience for him as well. Explaining even the simplest idea advances his understanding of the concept because he has to first think about how to explain it well enough to be able to convey the point. He also finds it rewarding, introducing students to whole new worlds of concepts and the fact that some of them are really receptive to these ideas is exciting to him. Speaking of interacting with students, he remembers fondly a class of about 30 biology students he had to teach when he held the post-doctoral position at the University of Chicago. He expected them to not be as motivated about a maths class they were forced to take, but he was impressed by the effort they put in, with almost 20 students crowding in his small office every week for office hours, “That was especially nice because they weren’t just doing a mathematics degree, so it was nice that they were so interested.”

“So, I like to do both. But I mean I wouldn’t have a career as just a teacher, that’s not what motivates me. What motivates me is to understand these problems, which I think are beautiful and that’s why every morning I’m excited to do mathematics, it’s because I want to solve these problems.”