You've done linear algebra, so what about solving equations that aren't linear? Algebraic geometry is the study of geometric objects coming from algebraic stuctures and using both geometry and algebra to understand the other better. Much like vector spaces are solutions to linear equations, we'll learn about how solutions to polynomial equations can be viewed geometrically and begin an exploration of affine varieties. We'll begin by getting a glimpse into what algebraic geometry is all about, and quickly realize that there's a lot of cool algebra that geometry elucidates. We'll encounter a little commutative algebra, some topology, and even some category theory! Time permitting, we may explore more advanced topics such as an introduction to curves.

Suggested References: Elementary Algebraic Geometry (Hulek, 2003)

Prerequisites: Math 31 or the equivalent

Representation theory takes the perspective that the right way to understand algebraic structures is to understand what they “do.” More precisely, a representation of a group is a vector space on which the group acts by linear transformations. This perspective has applications in physics and opens up a beautiful mathematical theory of its own. We will focus on the representation theory of finite groups and how it can be understood by looking at what are called “irreducible” representations, which act like atoms making up other representations.

Suggested References: Representations and Characters of Groups (James and Liebeck)

Prerequisites: Math 31 or the equivalent

How do people make decisions? How do people in a group interact? How and why do friendships form and break? These are just a few questions that can be answered using mathematical models. This project could take many different paths, depending on interest. We could study networks, game theory, evolutionary dynamics, voting systems, coordination, etc. Before starting, we will discuss what topic and text would be best for you.

Suggested References: Newman's Networks, Nowak's Evolutionary Dynamics, Arrow's Social Choice and Individual Values, etc

Prerequisites: Linear algebra recommended for some topics

A knot is an embedding of the circle into three-dimensions. It is just a tangled up circle sitting in space. Despite this simple description, it can be very hard to tell two such knots apart. Classically, mathematicians used easy to describe but hard to compute invariants of knots to tell them apart: crossing number, genus, unknotting number, slice genus, etc. Eventually, they progressed to associating polynomials to knots: the Alexander polynomial, Jones polynomial, HOMFLY polynomial, etc. The current state of knot theory is to associate even more complicated invariants, categorifications of these polynomials, to knots, to tell them apart, and in many cases, give us information about the more classical invariants. We can take a tour of these classical invariants, then through polynomials and onto categorifications. We will see lots of pictures and diagrams on the way.

Instead of considering pure math, we could also look for applications of knots, particularly to protein folding. There is a relatively new way of considering proteins in the language of knots, and we could learn about that and see what we can see. Whatever interests us the most.

Suggested References: The Knot Book (Adams), Knots and Links (Rolfsen), An Introduction to Knot Theory (Lickorish)

Prerequisites: Math 31 would probably be useful, and maybe Math 54, but don’t let those stand in the way of you learning interesting math.

Proofs are hard, and checking proofs can be taxing. The Curry-Howard isomorphism is a direct relationship between mathematical proofs and computer programs.

Suggested References: Lectures on the Curry-Howard Isomorphism (Sørensen and Urzyczyn)

Prerequitites: Math 31, or a proof based class; some computer science may be useful as well.