Projects 2023

Mathematical modelling of diabetes, with Katie Faulkner

What is mathematical biology? Sometimes called theoretical biology, mathematical biology is the use of mathematical models and tools to develop theoretical principles for biology. If we can translate known biological principles into some mathematical language, like an equation, then we can use the tools of mathematical analysis to discover new insights into the underlying biology.

Throughout this project, we will use one example biological system and one mathematical model of this system to introduce the field of mathematical biology and begin to explore the methods mathematicians use when studying biology.

Together, we will examine the Topp model of glucose regulation, a system of three ordinary differential equations (ODEs) that describe how your body keeps the concentration of sugar in your blood at a safe level. We will learn:

• how and why the authors chose the details of the model's form

• what this model can tell us about diabetes

• tools used in this paper that we can use to create and analyze new models

From here, you can pick up any other ODE model of a biological system and use your tools to learn something new!

Prerequisites: Single variable calculus (MATH 100 or equivalent), preferably with some exposure to differential equations. This project can be adjusted to the students' backgrounds depending on their level of experience with differential equations.

Arithmetic progressions and large sets, with Yuveshen Mooroogen

Suppose you're out for a walk at night, in your favourite part of the countryside. If the weather is nice, you'll look up and find a large, clear portion of the night sky filled with stars. It is reasonable that you should find some constellations among so many stars. You conjecture:

A large portion of the night sky should contain many constellations.

Unfortunately, things aren't always so simple. You may look up and find that some impertinent prankster has placed very thin clouds just where they need to be to block out your view of the stars! These clouds are small, so you're still looking at a large portion of the sky, but now this sky doesn't contain anything.

A large portion of the night sky may not contain many constellations.

In this project, we will study questions of the following form:

How large must a piece of the night sky be before it is forced to contain a constellation?

Questions like these are of great interest in number theory, combinatorics, and harmonic analysis. In this project, we will focus on problems concerning the existence of arithmetic progressions inside large subsets of the real numbers and the integers. 

First, we need to understand what "large" means. In the first part of the program, you will learn to work with several tools for quantifying the sizes of sets. We will start with elementary set-theoretic tools such as cardinality, then move on to analytic tools such as Jordan measure, topological tools such as density and thickness, and number-theoretic tools such as natural density. You will construct and study concrete examples to understand these sizes and the relationships (or lack thereof) between them. 

In the second part of the program, we will study two surprisingly accessible results about patterns in large sets. The first will be an old result from 1920, due to Steinhaus, and the second will be a modern result from 2019, due to Fraser. We will read the proofs of these results in detail and learn how they relate to the Erdős-Turán conjecture, a famous open problem from 1938 that has inspired much mathematical activity in recent years. Each week, I will assign problems to guide you through the papers and articles you will read, and we will meet to discuss them.

Prerequisites: One-variable calculus, including formal definitions of limits, infima (greatest lower bounds), suprema (least upper bounds), and convergence of infinite series. Ideal preparation would be either MATH 120 and 121, or MATH 226, or MATH 227. 

The Kakeya problem: in the plane and over finite fields, with Charlotte Trainor

Imagine there's a needle on a table in front of you (for our mathematical purposes, we'll pretend the needle has length one and width/thickness zero—just go with it!). You're tasked with rotating the needle 180 degrees, while covering the smallest area possible.

Certainly rotating through a circle of radius one would do the trick, but can we rotate the needle using less area? It turns out the answer to this question is yes, to an EXTREME degree. We can rotate it using arbitrarily small area—that is, if you give me a positive number epsilon, no matter how small it is, I can (theoretically, anyways) find a region of area at most epsilon through which I can rotate the needle!

This probably surprising and counterintuitive fact leads to one of the most important and still unsolved problems in math today: the Kakeya problem. As a stepping stone, mathematicians analyzed the analogous problem in a simpler setting—something called a "finite field." As the name suggests, finite fields have only finitely many elements, but can act as a very loose prototype for the real numbers. In 2008, a mathematician named Dvir found a surprisingly elegant and simple solution to the finite field Kakeya problem using polynomials.

 In this project, we'll discuss the following questions:

• How on earth can we rotate needles 180 degrees using essentially zero area?????

• What is the Kakeya problem? 

• What are these mysterious things called "finite fields," how can we set up the Kakeya problem in this setting, and how can we solve it using polynomials?

Prerequisites: Solid understanding of quantifiers and rigorous definitions of limits, and experience with proofs involving epsilons (MATH 120 would suffice). Familiarity with modular arithmetic, binomial coefficients, and multivariable polynomials would be useful, but the motivated student could catch up on these topics.