Before I officially started at UT Martin, I'd already said yes to my first undergraduate researcher. That summer, my senior colleague (later my faculty mentor) Dr. Curtis Kunkel mentioned a student, Larissa Renshaw, who had taken differential equations and numerical analysis, and asked if I'd want to work with her. I said yes without thinking twice. We started meeting over Zoom that summer, before I'd taught a single class here, and kept going through the year.
 

I'll be honest: I had no idea what I was doing. I handed Larissa pieces of my dissertation work and some code to modify, problems that were really beyond the scope of where she was, and I struggled to see where any of it would go. We met, we read, we tried things. In the end we put together a poster for the MAA-SE meeting in 2024, and called it a good first try.
 

But it left me with a question I couldn't put down: how do you actually do meaningful research with an undergraduate? My own work lives in functional analysis, which students can't reach in a semester or two. So in the summer of 2024 I sat with it, talked through ideas with Curtis, read his papers, and slowly an answer came into focus.

 

The problems I study in differential equations have discrete versions — the same questions asked on a grid of points instead of a smooth line. And those discrete problems are genuinely reachable. A student with Calculus 1 can work on the uniform discrete version of a problem I care about, another on the non-uniform version, and the mathematics is real, open, and publishable in undergraduate journals. That was the whole world of difference equations opening up, and it's where this program lives now.

 

The vision from here is simple: keep going in the same direction. I've worked with students on ordinary difference equations; the next step is partial difference equations — harder problems, the same idea, still within reach of a student who has taken Calculus 1.