# U of A Mathematics Associate Professor Chris Henderson Receives NSF CAREER Grant

Congratulations to Associate Professor Chris Henderson on receiving a National Science Foundation (NSF) CAREER grant. The Faculty Early Career Development Program (CAREER) is meant to jump start early career scientists “who have potential to serve as academic role models in research and education and to lead advances in the mission of their department or organization”. (nsf.gov) The grant will run through 2029 and it will support travel to conferences, visits to the U of A by other researchers, two undergraduate research projects, and the Applied Math Bootcamp. It will also support Chris’ graduate students “which is especially helpful for them so they can focus solely on their theses during the summer,” Chris shared. Learn more about the research projects from Chris below.

There are two big research projects under this award. The first has to do with reaction-diffusion equations and is focused on understanding precise long-time behavior. Reaction-diffusion equations model anything that involves diffusion (spreading) and reaction (growth). A quintessential example is that of an invasive species. Each individual wanders around exploring a new environment (spreading), while also reproducing (growth) up to the environmental carrying capacity (maximum population the amount of food can support). This can be at a microscopic level (bacteria like *e. coli*) or at a human (e.g., the cane toads in Australia). One can then augment these models to consider extra behavior of these species, e.g., the bacteria interact with each other via chemotaxis (they secrete a chemical that alerts their fellow bacteria to their presence and pushes them to aggregate).There are plenty of other physical systems that can be modeled by reaction-diffusion equations.

Our goal as a community is to understand what the universal behavior of these species is. For example, how fast do they invade a new environment? It turns out that the actual does not really affect this. Mathematically, we see that they invade at a fixed speed that only depends on physical assumptions like how quickly they reproduce and how much they spread out. Moreover, after “enough” time, the population distribution approximately takes the form of a “traveling wave.” My project looks to quantify this precisely using a new structure that my collaborators and I have uncovered in recent works. Basically, there is a hidden object, that we call the shape defect function, that quantifies how far away from the traveling wave the population distribution is at a given time. This object satisfies a nice reaction-diffusion equation itself, and I believe that by analyzing this secondary equation directly, we should be able to obtain interesting new results.

The second project is the Boltzmann equation, which models the distribution of particles in a dilute gas. Roughly, these particles move through space, constantly colliding with each other. This equation is notoriously difficult – it is “nonlinear” and “nonlocal” – so much so that it is even difficult to even understand if it is “well-posed.” In other words, theoretically, is there even a unique solution? Of course, we would never be able to write down a solution, but we could, e.g., approximate it numerically. Understanding well-posedness is important because it is related to our ability to use the model to make predictions. Generally, you would expect that if we knew everything about a cloud of particles (gas) right now, at this instant, we should be able to use the Boltzmann equations (which essentially encodes the laws of physics governing each particle) to predict what the cloud of particles will look like tomorrow. One such important cloud of particles for which the Boltzmann equation is the upper atmosphere, making it an important model in aerospace. That said, if unique solutions do not exist, we cannot expect anything useful from the model.

A measure of the difficulty of this problem is that, it has only very recently been solved in the much simpler case where the distribution of particles is space homogeneous; that is, the distribution of particles at every physical location looks exactly the same. This is obviously not at all realistic to the real world, but it was nonetheless a huge mathematical breakthrough! My project focuses on understanding what quantities control “blow-up,” which is essentially what would happen were well-posedness to fail. So far, in some simplified settings, we have shown that, unless one sees (say, numerically) a point of infinite mass or energy, the equation remains well-posed. There is a lot more to study, though. This result assumes that we are in a boundary-less setting, which is, of course, not realistic. We would like to understand how this is affected by things like the shape of the boundary (e.g., the confining walls of whatever is holding the gas).

Finally, let’s learn more about Applied Math Boot Camp. Chris shares, “The basic problem that I was trying to solve was that every year there are a handful of very bright Applied Math students that enter the program and struggle with the core classes because their undergraduate coursework was not in math or was at an institution with fewer class options. My goal was to identify such students and pay them to come on campus for a month before courses start in order to shore up their skills.

This summer was the first time that it ran. I had four incoming AM PhD student backgrounds work on a packet of problems, meeting with me for 3-6 hours a week. The goal was to get the students more comfortable with mathematical thinking and writing.

The packet of problems covers some basic point set topology and then constructs the real via Dedekind cuts. The goal was to find a topic interesting, if abstract, that would be totally new for all of the students. This way they would all be on an equal footing to start developing their mathematical reasoning and writing skills. The students who participated in the program this year were very enthusiastic about it, so I think it went well!”

Learn more about the research being funded in our department HERE.