• Proposed project: Optimal control and its applications

Description: Many complex systems can be modeled using a dynamical system. Phenomena in biology, physics, economics, marketing, and more can be described using differential equations. Oftentimes, we want to find the “best” outcome in any of these models subject to some constraints or goals. For example, what is the best amount of cash to keep on hand in order to maximize investments and meet cash demands? What is the best amount of medicine to dispense to a cancer patient in order to minimize side effects and tumor size? All of these questions and more can be answered using optimal control. In this project, we will learn the basics of optimal control and explore some of its applications.

Prerequisites: Differential equations, some programming experience in Python or Matlab would be nice, but not necessary.

• Proposed project: Mathematics and Redistricting

Description: Congressional districts are redesigned every ten years, a process with massive ramifications for the next decade's electoral dynamics. The drawing of these maps itself is an extremely politically charged process, and different maps can lead to vastly different outcomes. Most people have seen the wildly artificially shaped districts that are drawn "just so" to be able to maximize/minimize the voting power of one group on another. How can we tell if a given district map is fair? Are there any objective criteria for designing districts? What is the standard of evidence needed to argue that a map is not fair? It turns out these questions can all be framed with mathematics. We will study the work by Moon Duchin and the MGGG redistricting lab (https://mggg.org/) where they use anything from  Markov-chain Monte Carlo (to simulate "fair" maps) to topology (to characterize the "shape" of districts) to answer the above questions.

Prerequisites: Linear algebra and basic topology (i.e. intro real analysis) and at least an interest in statistics

• Proposed project: Don't Trust Neural Networks

Description: From image classification (cat or dog?) to Jeopardy! to Google searches, neural networks (NNs) have had tremendous success and continue to find utility across data science and scientific fields.  But can we trust them?  For instance, self-driving cars make decisions based on NNs -- we need to make sure we can trust those decisions.  In this project, we will learn about neural networks: the math behind them, the many applications for which they are used, and how easily they can be fooled.

Prerequisites: Multivariable calculus and linear algebra needed, some computer programming experience optional, but helpful.

• Proposed project: A Deep Dive into Matrix Analysis

Description: Matrix analysis is a fundamental topic in a wide range of applications, including data science, engineering, and machine learning. In this project, we will study advanced topics in matrix theory, focusing on more concrete and applied subjects (like eigenvalue perturbations, important matrix factorizations, positive matrices, etc.) than a typical linear algebra course. These topics are rarely covered in a typical undergraduate (or graduate) curriculum, but serve as the fundamental backbone of using linear algebra in practice.

Prerequisites: Linear algebra II (HMTH 375 for Morehouse, MATH 314 for Spelman) needed, numerical analysis useful but not needed.

• Proposed project: Investigating number systems through our understanding of the integers

Description: This project focuses on how our concrete experiences with mathematics help us build intuition, develop questions, and ultimately possibly answer them. We will investigate properties of the integers so that we can develop a lens to understand more abstract and intricate number systems that have many arithmetic properties that are similar to the counting numbers we know. This project is inquiry-based, so there are many avenues to study, mostly surrounding elementary number theory, algebraic number theory, but it would be open-ended based on the students interest.

Prerequisites: Experience with proofs, modular arithmetic, and base arithmetic (binary) is very useful, but not totally necessary.

• College: Morehouse

• Class year: '22

• Major(s): Mathematics and Computer Science

• Working with: John Urschel

• College: Morehouse

• Class year: '22

• Major(s): Mathematics

• Working with: Ashwin Narayan

• College: Spelman

• Class year: '22

• Major(s): Mathematics

• Working with: Ila Varma

• College: Morehouse

• Class year: '22

• Major(s): Mathematics and Engineering

• Working with: Elizabeth Newman

• College: Spelman

• Class year: '22

• Major(s): Mathematics

• Working with: Micah Henson