1. Burnside's Lemma: Using Group Theory to Count Distinct Objects

Burnside’s Lemma is a combinatorial result of group theory, which is often used when counting distinct mathematical objects up to symmetry. It is also known as Cauchy-Frobenius lemma, or "the lemma that is not Burnside’s, because Burnside is not the one who first came up with this lemma." This lemma was already known by Cauchy and Frobenius but was mistakenly attributed to Burnside when he stated and proved this lemma in his 1897 book Theory of groups of finite order without attribution. Burnside’s Lemma gives us a tool to count the number of orbits of objects, where two objects that are symmetric by some allowed permutations (for example, rotation or reflection) are categorized into the same orbit. First, we will study Burnside’s Lemma and the Orbit-Stabilizer Theorem with discussions of the relevant concepts in group theory. Then, we will use several illustrative examples to apply Burnside’s Lemma to count the number of orbits of objects with respect to symmetry. 

Prerequisites: Some proofs and a basic understanding of groups.

Reference: Topics in Algebra by Israel Herstein

2. Visual Graph Theory

A graph consists of a collection of points which we call nodes, and a collection of edges that connect these nodes together. An interesting example of a graph is the Facebook graph: A node on the Facebook graph is a Facebook user, and there is an edge connecting two different nodes (i.e. two different Facebook users) if the two users are Facebook friends. In this project, the student will learn the foundations of graph theory, and will learn to solve several amazing problems in graph theory, including the classic “seven bridges” problem, and the “theorem on friends and strangers”. 

Prerequisites: None.

Reference: Graph Theory with Applications by Bondy and Murty (link)

3. Two Limit Theorems in Probability

The law of large numbers and the central limit theorem are two fundamental theorems in probability. The law of large numbers shows how the sample means will converge to the actual mean, while the central limit theorem shows how they converge. In this project, the student will learn to prove one version of each of the two theorems, produce graphic demonstrations of the theorems, and apply them to a real world problem. 

Prerequisites: Multivariable Calculus, Linear Algebra.

Reference: Probability and Random Processes by Richard Durrett

5. Symmetry Groups

How many symmetries does a square have? We can reflect it across its diagonals and midlines, we can rotate it by 90, 180, or 270 degrees, and we can also combine these transformations to get even more symmetries! However, some of these symmetries end up being the same. For example, rotating the square clockwise by 90 degrees is the same as rotating it counterclockwise by 270 degrees. This makes counting symmetries difficult. However, the mathematics of groups and group actions allow us to solve many complicated problems like this one! In this project, the student will learn how to count, describe, and understand the symmetries of shapes using the theory of groups and their actions. 

Prerequisites: Calculus I, Calculus II, Linear Algebra. 

References: Groups and Symmetry by David Farmer

1. Random Walks and Electrical Networks

If you leave your house and walk around randomly, will you ever return home? In this project, the student will develop the basic properties of random walks. Random walks are stochastic processes that have application in modeling a number of complex, probabilistic systems. These walks can be understood by studying electric circuits. The student will explore the similarities in the properties of random walks on an infinite lattice, in one or multiple dimensions, and different types of electric circuits.

Prerequisites: Basic probability

Reference: Random walks and electric networks by Doyle and Snell (link)

2. Braid Groups 

If n is a positive integer, an n-braid is a collection of n strings that intertwine with one another. If we have two different n-braids, we can combine them by attaching the ends of the first braid to the beginning of the second. Under this operation, the collection of all braids forms the braid group B . The braid groups B are closely related to knot theory (Alexander’s theorem, Markov’s theorem), and also arise as the fundamental groups of certain manifolds, namely configuration spaces. In this project, the student will investigate the structure of the braid group, and how the braid groups arise in other areas of mathematics. 

Prerequisites: Abstract Algebra, Topology. 

References: Braid Groups by Kassel and Turaev

4. Introduction to Representation Theory

Groups and group actions are the algebraic gadgets that arise when one studies the symmetries of an object or space. They are very important and ubiquitous in math and physics, but they are quite hard to understand in general. A field of math we do understand very well is linear algebra. Often in math, we create tools to rewrite hard problems into simpler ones. Representation theory is a tool we use to take complicated math surrounding groups and simplify them into more approachable linear algebra problems. 

Prerequisites: Group theory

Reference: Groups and their representations by Karen Smith (link)

6. Fourier Techniques and Music

Fourier analysis is considered one of the crowned jewels of modern mathematical analysis and the introduction of Fourier techniques to has yielded manifold benefits. For instance, the Fourier transform's plays a major role in separating waves of different frequencies. Since chords and patterns in music are composed of different waves, the Fourier transform allows one to dissect musical chords with surgical precision. In this project, the student will learn the basics of Fourier analysis and write a program using the fast Fourier transform to apply their knowledge to their favorite musical chords.

Prerequisites: Linear Algebra, Multivariable Calculus, and some programming.

Reference: On the Mathematics of Music: From Chords to Fourier Analysis by Lenssen and Needell (link)

8. Riemann Surfaces, Monodromy and Polynomials 

We can think of the complex locus cut out by a polynomial in two variables as a number of sheets above the complex plane branched over a finite number of points. This locus is a two dimensional surface called the Riemann surface of the polynomial. There is a group naturally associated to this surface called its monodromy group. In this project, the student will learn to visualize and construct Riemann surfaces, define the monodromy group, and see how this group tells us about the solvability of polynomial equations. 

Prerequisites: Complex Analysis, Group Theory.

References: Algebraic Curves and Riemann Surfaces by Rick Miranda

9. Automorphisms of Sn

Groups often arise as symmetries of objects in math and nature, so it is natural to investigate their own symmetries. In this project we will focus on the symmetries of Sn, the symmetric group on n elements. First we will give an exposition on what the symmetric groups are and then prove that they are isomorphic to their automorphisms for most values of n. Finally, we will cover two seemingly unexpected constructions of the exceptional outer automorphism of S6.

Prerequisites: Linear Algebra.

Reference: Algebra by Michael Artin

2. Frobenius Elements in Galois Extensions 

In this project, the student studies the Frobenius elements with respect to some (finite) Galois extension of number fields. First they look at the general situation which is mostly understood by the celebrated Kummer-Dedekind theorem, which gives splitting information of prime ideals when extending to a larger number ring. Time permitting, they will use this to give a proof of the law of quadratic reciprocity. 

Prerequisites: Algebra 2

References: Lectures on Algebraic Number Theory by Yichao Tian (link)

3. Bordism of Manifolds 

A manifold is a topological space that locally looks like ordinary Euclidean space. We say that two manifolds M and N are bordant if there is a manifold W whose boundary is the disjoint union of M and N. Bordism classes of manifolds can be given the structure of a ring, where M + N is the disjoint union of M and N, and M × N is the cartesian product of M and N. In this project, the student will work towards Thom’s calculation of the (unoriented) bordism ring. Along the way, the student will encounter many important elements of modern algebraic topology, such as spectra and generalized (co)homology theories. 

Prerequisites: Algebraic Topology (e.g. Chapters 0, 1, 2 of Hatcher’s Algebraic Topology), familiarity with smooth manifolds and transversality, familiarity with groups, rings, and modules. 

References: Bordism: Old and New by Daniel Freed (link) and Bordism Homolgy Lecture by Pierre Albin (link)