# Research Spotlight: Victor Barranca and Applied Mathematics (Part II)

*This is the ninth interview in the series Research Spotlight, in which I share conversations that I have with faculty regarding their research, their journeys within their fields, and their fields in a broader context. This is Part 2 of this article. Read Part 1, here.*

*Victor Barranca is an assistant professor in the Department of Mathematics and Statistics.*

Aidan Reddy: What are some philosophical questions or thoughts that you have pertaining to your research?

Victor Barranca: One philosophical problem that commonly arises is that, when you are going to construct a mathematical model, you generally can’t, with full accuracy, describe every single thing that is happening in a neuronal network. You have to make certain choices, certain simplifications. One question that folks will often pose is, “Can I still learn something meaningful, even though I have abstracted in some sort of way?” In my view, the answer is “yes”, in the sense that, often by making specific idealized assumptions, we can discern specific mechanisms that would otherwise not be revealed. If our model is too complicated, we might not be able to focus on key features that reveal to us a particular mechanism in cognition. There’s always this tradeoff between level of model realism and analytical tractability. Overall, if we choose the right aspects to focus on, we can come up with very meaningful answers to concrete scientific questions in a very mathematically-elegant way. Making this philosophical choice is central to the art and science of applied mathematics.

Of course, there are other big-picture questions, like, “Can I explain cognition fully just based on neuronal dynamics?” Part of being in mathematical neuroscience is taking the viewpoint that the answer is “yes” — that is, cognition is a direct consequence of what neurons are doing. That’s the avenue that I take at least when I do my own research.

AR: How did you become interested in math, and, specifically, applied math and mathematical neuroscience?

VB: It was sort of a long road, but looking back, I think the pieces fit together nicely. I first became interested in math late in high school. Early on, I had pretty diverse academic interests and no real familial ties to math. So, it wasn’t until I took calculus in my last year of high school that I started to become excited by math. In particular, I was taking physics at the same time, and I noticed these really interesting connections between what we were doing in calculus and in physics — that you could express physical laws really elegantly using math. My math teacher that year was great. He would come up with super creative lesson plans, involving things ranging from dressing up as Isaac Newton to setting up a mathematical crime scene and letting us solve it. That year sparked a genuine interest in math, so ultimately I decided I’d go on to major in it. It wasn’t really until a few years into the major that I understood the breadth of what higher level math involved. I enjoyed the standard classes in math from multivariable calculus to real analysis, but, the class that really struck me was foundations in applied math, which was the first time I saw applications of math primarily focused on areas besides physics. If you look at a standard calculus book, 90% of the applications are probably from physics. This was one of the first classes where I saw applications in chemistry and biology. I saw a concrete methodology used to solve real-world problems across the natural sciences. I was also a psychology minor as an undergrad, so when I learned that I could combine ideas from neuroscience and mathematics, I was sort of sold.

AR: What does the future of applied math look like?

VB: That’s a tough question. The landscape is constantly changing. If you asked me this question a few hundred years ago, the answer would have largely involved physics. It would have involved describing the dynamics of visible objects. Later, scientists developed mathematical tools to help describe dynamics at a quantum scale. In the early days of applied math, most efforts were directed towards physics and then engineering. As time progressed, applied math became increasingly used in new disciplines, such as economics and computer science. As the sciences became more interdisciplinary, important mathematical connections, such as to biological systems, were increasingly made. Similarly, when novel network science tools were developed, quite recently, then mathematics became applied to describe, for example, large-scale interactions. So, applied math, really, advances along with the current state of the sciences. Right now, we have all of this data from our increasingly digitized world, so a big part of applied math currently is data science — formulating mathematical mechanisms to draw insights from data. In relation to my field, now that we’ve been able to explain brain function at a finer scale and we have more brain data, neuroscience has become much more mathematical. The life sciences, in general, have become more mathematical. So, at least in the next twenty years or so, I expect a large outflow of mathematical findings related to biology, especially since many biological systems lack a unified mathematical description. This work is especially difficult because these systems are exceptionally complicated and have many different types of math describing what’s going on. So, coming up with nice way to reconcile all of these areas of math with biological observations will likely be a key area of investigation.

AR: What would you say to someone who feels intimidated by math, or that math isn’t for them, for any reason it may be?

VB: This question comes up a lot. I think part of the solution to dealing with this issue in mathematics is acknowledging that, just like any other worthwhile craft, mathematics can be difficult if not conveyed effectively. In my view, the problem can be solved by A: dispelling any mathematical anxiety stemming from earlier mathematical difficulties and B: giving the right motivation for doing mathematics. Often times, when I talk to someone about their difficulties in, say, calculus, it’s not so much the new mathematical ideas that they’re having a hard time with; it is the fundamentals — the algebra, the trigonometry. So, I think part of the solution is meeting a student at the right mathematical starting point, and making sure that they have the fundamentals down before they move on to something more complicated. Sometimes this involves explaining concepts in a creative manner and other times imparting the right intuition as ideas grow in complexity. As mathematics becomes more abstract, I find it’s helpful to make sure that students have the right intuition to grab onto through these murkier waters. If a student can hold onto an illustrative example or big picture story, they can move forward uninhibited. Once students feel confident in their mathematical ability, then they can make strides in learning new mathematics, and become excited rather than intimidated by new mathematics.

A counterpart to that is convincing someone that math is worth doing — that math is powerful. Us mathematicians often view math as elegant and beautiful, and thus worth doing. But a lot of people don’t necessarily have that perspective and need more tangible motivations. That’s perfectly understandable. What you should instead convince them of is that math is powerful — that it’s a tool useful either in their daily lives or rather to society as a whole. Sometimes, it’s just finding that right application, the right spark that really gets a person interested in math and convinced that they should put their effort into learning it. When I teach mathematics, I apply this philosophy by discussing a broad array of applications and also imparting a sense of the big picture of the mathematical framework built over the entirety of a course. Just like art or literature, mathematics is diverse, and we all might have different tastes in what is mathematically interesting, but you can almost always find at least one aspect of mathematics well worth exploring.