• Proposed project: A Study in Chaos: Exploring the Dynamics of Disorder

Description: At first glance, the world around us can seem orderly and predictable. After all, the laws of mathematics and physics are clear and consistent. So, why is it so hard to accurately predict things like the weather?

A dynamical system is anything that changes over time according to a specific set of rules. When we zoom into the study of certain systems, we encounter a fascinating paradox: even simple systems can exhibit sensitivity to initial conditions, leading to long-term unpredictability. Chaos ensues, as explained in this scene from the classic Jurassic Park movie: https://www.youtube.com/watch?v=3lZy3teNY84.

In this project we will study such randomness in apparently deterministic systems. On our quest, we will journey through the basic principles of nonlinear dynamical systems and chaos, including stability, bifurcations, fractals, and strange attractors. This involves understanding how systems respond to small changes and whether they return to their original state or evolve differently, how systems can undergo sudden and dramatic shifts in behavior, and how complex, seemingly random patterns can characterize the long-term behavior of chaotic systems. We will do so with an eye towards applications. 

You can witness an example of chaos unfolding here (“Butterfly effect”): https://www.youtube.com/watch?v=oqDQwEvHGfE

The project can be modified to meet the student's math background and interests. 

Suggested Books: "Nonlinear Dynamics and Chaos" by Steven Strogatz, and "Chaos: An Introduction to Dynamical Systems" by Kathleen Alligood, Tim Sauer, and James Yorke

Prerequisites: The essential background needed is single-variable calculus. In a few places, multivariable calculus and linear algebra may be required, but we can spend time understanding the topics we need from these areas as we go.

• Proposed project: The Many Faces of Geometry

Description: What more is there to geometry than what you learned in high school? In this project we’ll explore four different perspectives of geometry: the Euclidean construction, linear algebra, projective geometry, and transformation groups. Each approach offers a unique view and builds on what came before, revealing how versatile geometry can be.

We will begin with the foundations laid by Euclid and progress to modern techniques like Lie groups and hyperbolic geometry. Along the way, we’ll connect geometry to other areas of mathematics like algebra. At the end, you'll be able to build geometric intuition from multiple "angles" (pun intended)!

Suggested Book: "The Four Pillars of Geometry" by John Stillwell

Prerequisites: Calculus 2, and a proof-based math class would be helpful

• Proposed project: Applications of Mathematics: Network Theory in the Real World

Description: At times it may seem difficult to see how mathematics applies to the real world. Through the exploration of Six Degrees: The Science of a Connected Age by Duncan J. Watts, we will dive into the world of network analysis and see how graph theory can help us understand real world situations. The student may decide what aspects of graph theory or network analysis to focus on.

Prerequisites: Basic knowledge of graph theory

• Proposed project: Election Year Mathematics: How Much Does Your Vote Count?

Description: Partisan gerrymandering is the practice of drawing electoral district boundaries in such a way that advantages a particular political party. This practice can lead to more seats (hence more political power) being awarded to a party than its share of the vote would suggest. But how do we prove that a district has been gerrymandered? Over the years, mathematical tools have been developed to detect partisan gerrymandering. For this project, we will first work to understand some of the history of partisan gerrymandering and the social and political ramifications thereof. Then we will take a look at a metric designed to gauge partisan gerrymandering, the efficiency gap, and discuss its uses and shortcomings. We will also learn about the so-called "efficiency principle" and alternative metrics to the efficiency gap.

Suggested Paper: "The Efficiency Gap, Voter Turnout, and the Efficiency Principle" by Ellen Veomett

Prerequisites: Basic course in linear algebra

• Proposed project: An Introduction to Chip-Firing

Description: Chip-firing is a beautiful and relatively modern topic in combinatorics that imagines placing "poker chips" at the vertices of a graph, and then allowing them to "fire" to adjacent vertices. This simple idea can be used to create beautiful fractals, understand the algebra around spanning trees, find discrete analogues of some algebraic geometry concepts, give a model for systems like avalanches and earthquakes, and so much more! One of the best aspects of this topic is that there are many different avenues to explore, and the student can choose what appeals to them the most.

Suggested Book: "The Mathematics of Chip-Firing" by Caroline Klivans

Prerequisites: Very flexible, but some linear algebra would be a plus.

• College: Spelman

• Class year: '25

• Major(s): Mathematics-Civil Engineering dual degree, Environmental Studies Minor

• Working with: Millicent Grant

• College: Morehouse

• Class year: '26

• Major(s): Mathematics

• Working with: Sam Freedman

• College: Morehouse

• Class year: '27

• Major(s): Mathematics, Chinese Studies Minor

• Working with: Alex McDonough

• College: Spelman

• Class year: '26

• Major(s): Mathematics

• Working with: Emille Lawrence

• College: Spelman

• Class year: '26

• Major(s): Mathematics

• Working with: Jasmine Bhullar