Presentations are Friday, December 5, 2025 from 12:30pm-3:00pm in SEO 636!
(mentored by Hassan Babaei)
In this project, we begin by reviewing introductory ideas from quantum mechanics, focusing in particular on the Heisenberg Uncertainty Principle and its interpretation of position–momentum measurements. We then introduce the Fourier transform as a mathematical tool for analyzing functions in both physical and frequency space. By exploring its basic properties such as inversion, scaling, convolution, and examining how it interacts with translations and modulations, we show how the Fourier transform naturally leads to a precise mathematical formulation of uncertainty principle.
(mentored by Darius Alizadeh)
Mixing is an important concept in ergodic theory but to measure mixing we need to understand how to measure sets. I'll define the Lebesgue measure and mixing in order to understand some examples of sets and systems.
(mentored by Katie Kruzan)
Quantum Algorithms offer faster solutions to certain classical computing problems by taking advantage of quantum properties such as super positioning. Using the quantum circuit model we are able to better understand what is necessary for their implementation and the limitations to the benifits that they provide.
(mentored by Clay Mizgerd)
We will discuss the evolution of random graphs.
(mentored by Jennifer Varcaro)
Hyperbolic space differs from Euclidean space in that its triangles haev interior angles less than 180°. In this talk, we will introduce geodesics and quasi-geodesics, state the Morse Lemma, along with an interesting application if time permits.
(mentored by Theo Sandstrom)
One of the most useful properties of a metric space is the notion of completion, or the property that all Cauchy sequences in a metric space converge to a point also in such space. We will show that all metric spaces have a completion and give an outline of what the process of completing a space looks like. We will conclude by observing a useful completion of the rational numbers, namely the p-adic completion.
(mentored by Michael Lange)
This presentation gives motivation of the Lebesgue integral by introducing and constructing Lebesgue measure, we first introduce the Riemann integral and point out its limitations, then we explore how Lebesgue measure overcomes these issues. This motivates restricting our attention to σ-algebras, leading to the Borel sets and ultimately to the Lebesgue measurable sets. We conclude the talk by discussing interesting examples from the Cantor set.
(mentored by Lisa Cenek)
This presentation offers an introductory overview of knot theory following Colin C. Adams’ The Knot Book. We begin with the formal definition of a knot and examine how Reidemeister moves provide a foundation for understanding knot equivalence. We then explore tricolorability as an accessible yet powerful knot invariant, followed by a look at the tabulation of knots and the methods used to classify them. The talk concludes with a brief introduction to the relationships between knots, links, and graphs. By closely following Adams’ exposition, the presentation aims to build clear intuition for the central ideas and tools of modern knot theory.
(mentored by Ping Wan)
Group theory has been used as a tool in mathematical virology due to the viral capsids' symmetry properties, especially icosahedral symmetry. Taking inspiration from the classic Caspar-Klug theory and the cut-and-project method used in the construction of the Penrose tiling, it turns out we may know a little more (and less) about these tiny mysterious entities than we think.