➤ Links, Lattices, and Rational Homology 4-balls

Advisor: Jon Simone

Description: The study of knots is an important field within topology. For example, one can use knots to draw blueprints for 4-dimensional manifolds. Hence understanding particular properties of knots (and collections of knots called links) allows us to better understand 4-dimensional objects. During the summer, students will: explore a particular geometric property of knots called sliceness; learn how linear algebra (in the form of lattices) can be used as a tool to understand this geometric property; and see how slice knots can be used to construct special 4-dimensional manifolds called rational homology 4-balls. Throughout the summer, students will also work on an original project related to these ideas.

➤ Radio Labelings of Graphs

Advisor: Aleyah Dawkins

Description: First introduced in 1980 by Hale, the radio assignment problem considers assigning frequencies to transmitters in a network in such a way that interference is avoided and the spectrum of the transmitters is used efficiently. For a simple connected graph $G$, a radio labeling of $G$ is a function $f:V(G)\to\mathbb{Z}_+\cup\{0\}$ that satisfies $$\begin{equation}\label{radiocondition} |f(u)-f(v)|\geq\text{diam}(G)+1-d(u,v) \end{equation}$$ for all distinct vertices $u,v\in V(G)$. We will study graphs that offer a solution to the problem through a radio labeling using various techniques, in efforts to characterize when a graph admits such a solution.

➤ Modeling liquid crystals with numerical PDE's

Advisor: Andrew Hicks

Description: Liquid crystals (LC's) are a material that has increasingly found many uses in our modern world, most famously in liquid crystal displays. This research looks into the behaviors of LC's through the lens of numerical PDE's, in particular using the finite element method (FEM). The model used is the Landau-de Gennes (LdG) model, which uses an object called the Q-tensor to study the behavior of LC's. This project will study new aspects such as reproducing real-life LC laboratory experiments via simulation, or doing simulations on the optimal control problem for LdG.

Prerequisites: A course in differential equations and some programming skills, preferably in Python.

➤ Using LEAN to teach proofs

Advisor: Pavel Kovalev

Description: The topic of interactive theorem proving has been receiving increasing attention from the mathematical community. Most efforts have focused on digitizing mathematics and verifying the correctness of proofs. Less attention has been given to using Lean for pedagogical purposes. At CMU, there is a course on interactive theorem proving that assumes students are already familiar with pen-and-paper proofs, which are taught in a separate course. An interesting enterprise would be to reverse this order, using Lean as an educational tool to teach students basic proof techniques. This project aims to explore exactly this idea -- incorporating Lean into the teaching of an introduction to proofs course in a meaningful way. Students will learn to use Lean and work on formalizing the content of the standard CMU introduction to proofs course in a way suitable for educational purposes.