Million Dollar Mathematics

Course Description

Although not all open problems in mathematics come with a million dollar prize, some definitely do! Math is a changing and growing subject and new discoveries are being made all the time. Through experimentation and with the help of computers, we will discover how research mathematicians draw their conclusions. If we are lucky, we may even solve a million dollar problem!

This course is an overview of the process of mathematical research, from forming the hypothesis to computational tools and types of proofs. We will begin by providing an introduction to Matlab, a programming environment used frequently by scientists and mathematicians. Then we will use computational techniques, hands-on experiments, and intuition to answer questions in various fields of mathematics as one would do in an actual research environment.

We will discuss, among others, prime numbers and their properties ("How far apart are prime numbers on average?", "How many Fibonacci numbers are prime?", etc.), applications of probability to real world situations ("What does randomness mean?", "Who is more likely to win at a card game?", "Can we use probability to detect tax fraud?", etc.), models for infectious diseases, and fair division in various contexts ("What is the best way to assign Congress seats?", "What is the most fair way to share rent?", etc.). Students will be invited to consider these questions and use their intuition, as well as numerical methods, to suggest an answer. Students are encouraged to propose their own questions to investigate. Depending on their interests, we will explore more advanced topics, such as arithmetic dynamics or hidden Markov models.

By the end of this class, students will understand what a career in mathematics entails and the types of problems mathematicians consider. Just as importantly, they will become comfortable with a programming language that is widely used both within and outside academia.

Prerequisites: There are no prerequisites for this course other than an interest in mathematics, coding, or research in general. We will cover every concept in detail, including the programming component, and we will assume students have no previous coding background.