Summer 2025 Projects
- Math 398
- 5 credit, Credit/No Credit grading
- Independent study research course meeting with faculty weekly
- Does not count towards math degree requirements
Project A: The Teaching of Mathematics
Instructors: Jim King and Charles Camacho
As a student, you have experienced a lot of math instruction. This course is an opportunity to explore what goes into math teaching. The course will provide examples and activities at a variety of levels and topics to expand your knowledge and stimulate your imagination. Then you will create a project on a topic you choose to pursue inside this big subject.
Prerequisite: None
Meets remotely, M/W 2:30-4:00, message board for discussion between meetings.
Project B: Calculus Based Projects
Instructors: Charles Camacho and Andy Loveless
Have you ever wondered how calculus is used to understand the world around us? In this course, you will explore engaging calculus-based projects from single-variable differential and integral calculus and even create one of your own! Projects include playground mathematics, cutting pancakes in higher dimensions, and the mathematics of tennis. Students will also work closely with a faculty member to apply calculus to a project that they design.
Prerequisite: MATH 125 Calculus II
Meets remotely, Tuesdays 1-3pm
Project C: Approximating Algebraic Structures
Instructor: Be'eri Greenfeld
This project will explore a class of questions arising in asymptotic algebra, such as: Can we find matrices A and B such that AB−BA=I? Can we find matrices for which AB−BA is “close” (in some sense) to the identity matrix I? Given matrices A and B that almost commute (i.e., AB≈BA), can we perturb them slightly to obtain new matrices A′,B′ that do commute? The answers to these questions are (respectively): no, yes, and sometimes. These problems lie at the intersection of several active areas of contemporary mathematical research, including metric algebra, quantum information theory, and approximation theory in group and operator algebras. The primary aim of this project is to develop an understanding of the theoretical foundations and current research surrounding approximations of algebraic structures.
Prerequisite: MATH 403 Intro. to Modern Algebra - Elementary Theory of Groups
Meets in-person, M/TH 10:30-noon
Project D: A Markov Chain Model of Social Media Polarization
Instructor: Soumik Pal
It is often said that social media leads to polarization in society and the algorithms used by social media companies have an immense impact on the level and speed of polarization. This project will aim to build Markov chain models of social media to display the effect of policy on polarization. We also aim to infer policies that can reverse polarization with minimum cost. The goal is to write a project report with graphics made for presentation.
Prerequisites: undergraduate probability, Python programming, and knowledge of Markov chains is preferred
Meets in-person, MON 11:00
Project E: Exploring Super Vector Spaces
Instructor: Guillermo Sanmarco
In order to understand an algebraic object (such as a group or an algebra), it is customary to study linear actions of said object in vector spaces over a fixed field. By doing so, we can use the well-understood theory of vector spaces to get a better idea of our original object. One of the guiding principles in modern algebra suggests that we can obtain more information by studying the collection of all such linear actions. In fact, this newly formed collection usually possesses very rich structures of its own. The study of these structures becomes of independent interest, which is precisely the goal of the theory of Tensor categories.
Prerequisite: MATH 318 or MATH 336 or MATH 340 or MATH 402
Meets remotely, WED 10:00 to noon
Project F: Beyond the fourth dimension
Instructor: Sándor Kovács
It is often said that time is the fourth dimension. In fact, in the Minkowski spacetime four dimensional model the three spatial and one time dimensions are considered equivalent. More recent research, string theory, suggests that our space is even higher dimensional, it is at least 10 (superstring theory), or possibly even 11 (M-theory) or 26 (bosonic string theory) dimensional. The reason we are not aware of these extra dimensions in our everyday life is that they are extremely tiny. Imagine that you were confined between two walls and not able to move your head toward the wall. You would only perceive a two-dimensional world even though your body is three-dimensional. String theory suggests that our 4-dimensional spacetime is confined between these very high dimensional walls that are extremely close to each other.
The goal of this project is to understand the precise mathematical description of these higher dimensional models. An interesting feature is that these extra dimensions are not linear which makes their geometry more complicated and simultaneously much more interesting.
We will aim to get to a level of understanding that will allow discussing current open mathematical problems in the area.
Prerequisite: a minimum 2.0 in Math 318 or Math 402
Meets remotely on Tuesdays 10-noon