Current Projects

Mentor: Vincent Solon

Area: Functional Programming & Category Theory

Overview of the topic: "Haskell is a lazy, statically-typed, functional programming language. Many people who use Haskell seem to also like category theory, and many of the core programming patterns use categorical ideas. To actually learn Haskell, there are several resources. I learned a lot from a textbook, but there are online guides such as https://learnyouahaskell.github.io. To learn relevant category theory, we can read Vakil chapter 1 if you have somewhat of a background in algebra. There are other books and notes on pure category theory, as well as category theory with a functional bend as well. As an end goal, on the applied side of things, there are some fun Advent of Code problems we could do (though the Christmas theme may be off-putting). On the theoretical side, we could learn the Curry-Howard correspondence, which provides a correspondence between computer programs and mathematical proofs.

Pre-requisites: It would be nice if you have written code before, in some language, but it doesn't seem necessary to me. If you have some background in rings/modules, it would make Vakil chapter 1 seem more motivated and a better choice.

Text: "Vakil ""The Rising Sea"" Chapter 1 (a pdf is available online), https://learnyouahaskell.github.io  or ""Programming in Haskell"" by Graham Hutton"

Mentor: Nathan Bushman

Area: A Sampler of Number Theory

Overview of the topic: We will survey a variety of classical topics in number theory. A selection of those topics, according to student interest, will be explored in more depth.

Pre-requisites: Student should not have taken MATH 4330 already. Otherwise, no prerequisites in particular: project is open-ended, and could be adapted to either pre- or post-MATH 3000 students.

Text: "A Friendly Introduction to Number Theory," Joseph H. Silverman

Mentor: Brent Koogler

Area: Control theory

Overview of the topic: We would begin with an introduction to linear state-space systems, and then we would choose topics based on the mentee's interests. Possible topics include nonlinear dynamical systems, optimal control, models of the vestibular system, and model predictive control. These topics develop applied and/or theoretical skills, e.g., computer programming and/or theory on L^p spaces.

Pre-requisites: You should have taken the calc sequence, and you should understand what a differential equation is. The other advanced topics might require more prerequisites, but we can study that material as needed.

Text: Linear state-space control systems / Robert L. Williams II, Douglas A. Lawrence

Mentor: Arun Suresh

Area: Metric Geometry 

Overview of the topic: Suppose there is a collection of points X in the n-dimensional Euclidean space R^n (you don't know what the points are). Now, suppose I tell you all the inter-point distances between the points (that is, you know d(i,j) = |p_i - p_j|^2). The question is: 

• Can you recover X from just this data? 

• What if I say the distances I give you are noisy? 

• What if I remove all the labeling information (in this case, you are given all the distance numbers without knowing which distance goes with which two points)? Can you still do it? 

In this project, we will consider this problem and explore some existing work towards solving it. We will also study some applications of this problem to areas of acoustic vision (reconstructing shapes from echoes). Time permitting, we can explore a slight generalization of this problem. What if X is not Euclidean? What if X is a curvy space with a different notion of distance? This might touch on reading some original research work to some extent. 

Pre-requisites: You must have taken matrix theory (or know how to work with matrices), having taken abstract linear would be even better. 

Text: "We will start with this very well-rounded and beginner friendly expository paper: https://arxiv.org/pdf/1502.07541. We can also look at this book: https://www.convexoptimization.com/TOOLS/0976401304_v2010.08.25.pdf Based on student interest, I can make other resources available"