• Proposed project: Solving differential equations by approximation

Description: Many real-world applications are modeled using ordinary differential equations (ODEs) – even in complex systems involving partial differential equations, solving an ODE is a critical step. It is often impractical or impossible to find analytic solutions by hand. In practice, we rely on approximating solutions to ODEs using numerical methods. Approximation isn't always a comfortable concept to a mathematician, but it is one of the most important tools in an applied mathematician's toolbox! In this project, we will learn how to derive some standard numerical methods for solving ODEs. We will also focus on how to evaluate a given method, asking questions about its accuracy, its stability, and examining situations in which the method may fall short. Depending on the student's interest, we can pursue a more theoretical approach with derivations and simple direct proofs, or we can introduce some coding to actually implement the methods we learn about.

Suggested Book: "Numerical Analysis" by Tim Sauer (Chapter 6)

Prerequisites: Calculus 1 and 2. A course in ODEs may be helpful for background information, but is not necessary.

• Proposed project: Matroids: A Geometric Introduction

Description: Matroids are powerful mathematical objects that generalize key concepts from linear algebra, like linear independence, relations, rank, and bases, to other fields like graph theory and combinatorics. They have applications to optimization, coding theory, and combinatorial geometry, and are used to solve problems involving structure, independence, and efficiency. In this project, we will start by studying matroids from the perspective of graphs, focusing on the various ways to define and construct them, and then look at them from the lens of unusual geometries such as the Fano plane. From there, we can explore various applications that interest us!

Suggested Book: “Matroids: A Geometric Introduction” by Gordan and McNulty

Prerequisites: Linear algebra and some exposure to proofs. Some basic knowledge of graph theory would be nice, but is not required.

• Proposed project: Introduction to Machine Learning

Description: At one point or another, we have all heard the term "Machine Learning,” especially now with the rise of AI. But what exactly is Machine Learning? The goal of this project is to build a precise understanding of this. Although the field is so vast that it's not possible to go over all its aspects, we will look at some of the foundational theory and algorithms that shaped the field's development. Our goals include: Define and understand the different areas of Machine Learning. Deconstruct key algorithms (Regression, K-Means, Neural Networks) and understand the mathematical intuition behind them. If time permits, we will try to code some of these algorithms.

Suggested Book: We will mostly follow "Machine Learning for Absolute Beginners" by Oliver Theobald, and I will provide extra references as well.

Prerequisites: Basic Calculus and Linear Algebra. Coding experience is not required.