For all the readers, who are you, what do you teach, and what is your area of expertise?

I’m Victor Barranca. I teach classes across mathematics, especially in applied math. I do research in nonlinear network dynamics and mathematical neuroscience.

What classes are you teaching this semester, and what’s your favorite class to teach (and why?)

This semester, I’m teaching Stochastic and Numerical Methods, our core upper-level applied mathematics course. My favorite class to teach is probably Modeling, since it’s uniquely structured so that applications directly motivate the development of new mathematics. It allows us to also explore a diverse range of applications, so that at least one or two will almost surely resonate with every student. Plus, it’s a great class for introducing students to my research area.

So, what first drew you to math, and what keeps you interested in it?

Calculus was the class that initially drew me to mathematics, and foundations of applied mathematics was the class that kept me hooked for many of the same reasons I enjoy teaching the modeling course. Games and puzzles have always appealed to me, so the problem-solving and playful side of mathematics definitely keeps me interested. The human brain is, in my view, one of the most complex systems, and its many mysteries are a rich source of diverse mathematical problems that always have me thinking about new research questions.

What do you want students to get out of your class?

I hope students in my courses ultimately see that mathematics can be three things at once: beautiful, impactful, and fun! Teaching mathematics is in many ways like telling a good story. I believe there needs to be an interesting and clear narrative, which connects all the crucial ideas of a single class meeting, equipped with intrigue, suspense, and discovery, like any good story. Many of my courses also strive to blend rigorous and theoretical questions, which can have surprisingly elegant, yet insightful, solutions, with application-driven problems, which may instead require modeling, approximations, and technical calculations. My aim is that this spectrum of mathematical ideas appeals to a broad swath of students in an approachable and exciting way.

What’s a fun mathematical concept or idea a student has introduced you to?

In many of my upper-level applied math classes, students perform an application reflection towards the end of the semester. These reflections are meant to give students a chance to find direct applications of the course material not discussed in class meetings and share them with their classmates in a blog post. My hope is that this provides students with a low-stakes opportunity to explore an area of application that’s personally important to them. Since the students are required to respond to each other’s posts, this also exposes students to mathematical ideas they may not have ever thought of beforehand too. Needless to say, I have encountered many interesting and unfamiliar topics in these posts each semester, recently including applications like slime mold behavior and solar heating systems.

What’s a mathematical concept or idea that you think is really cool, and you can summarize here in a few sentences?

This is a recent research question that I think is really cool, fast to describe, and surprising. While our brain may be optimized to process natural stimuli, surprising deficiencies occur for non-natural stimuli. Typically, there is a lot of correlation in the light entering each of our eyes, imparting many benefits, like depth perception. However, consider an experiment where the left and right eyes are simultaneously presented with very different images. What do we perceive and why? What generally occurs is that we perceive just one of the two monocular stimuli for a while and then we randomly begin to perceive the other, with random alternations in percept as time goes on. Since the stimulus is constant while the percept is dynamic, this phenomenon must be rooted in the structure of the visual system and can tell us much about the structure-function relationship in the brain. Mathematics allows us to systematically explore potential mechanisms for this behavior and to leverage it in unexpected directions, such as towards understanding certain neurological disorders. Going full circle, I see beauty, impact, and enjoyment of mathematics in this problem!

Thank you so much!

Victor Barranca is an Associate Professor at Swarthmore College, and, if the author may say so, a wonderful differential equations professor.

Images taken from Swarthmore College website and XKCD, with a CC-2.5 license.