• Proposed project: Combinatorial Game Theory

Description: In this project, we will explore the theory of combinatorial games, which are games between two people who have perfect information and do not end in a draw.  The study of combinatorial games was revolutionized by Conway, who introduced the surreal numbers. Surreal numbers can be understood as games themselves and can be used to classify and study games. We will discuss impartial games (such as Nim, where players have the same available move sets and win conditions) and partisan games (such as Go, where players can have different move sets and win conditions). We will explore ways to find and prove winning strategies (e.g., strategy stealing), but also the general theory of winning strategies. For example, we can show that in combinatorial games that end in a finite number of turns, there must be a player with a winning strategy. We can also show that every impartial game is essentially a game of Nim, whose winning strategy is well understood (the Sprague-Grundy theorem). After discussing examples and elements of the theory, we have the option of exploring concrete applications to the game of Go.

Suggested Book: "Combinatorial Game Theory" by Siegel

Prerequisites: At the minimum, students should have an exposure to basic proof techniques and set theory.  Basic knowledge of graphs is helpful.  Any other background will be introduced in the course.

• Proposed project: Keeping (or Decoding) Secrets — the Mathematics of Cryptography

Description: Code making and breaking is a cornerstone of internet security — running in the background of our daily lives to protect valuable information. The fundamentals that allow such ideas to work are purely mathematical and in constant development. In this project, we aim to develop a working understanding of how to encrypt and decrypt messages using mathematical algorithms, with a focus on transitioning from historical roots to the cutting-edge ideas used today. Early topics will include ciphers from antiquity up through the Enigma cipher— whose use famously fueled WWII and whose decryption helped end it.

Later lessons are flexible in terms of direction, which may include topics such as elliptic curve cryptography, quantum cryptography, and error-correcting codes. Along the way, students will see the fundamentals of abstract algebra, combinatorics, number theory, and complex analysis — no prior experience required (though this will allow for much deeper investigation!).

Suggested Book: "Protecting Information: From Classical Error Correction to Quantum Cryptography" by Susan Loepp and William K. Wootters

Prerequisites: Good understanding of linear algebra is needed, and a proof-based math course is strongly preferred. Any of the following areas: algebra, combinatorics, or analysis would be a plus but not necessary!

• Proposed project: Representation theory of finite groups and applications to quantum mechanics

Description: The representation theory of a group G describes all the ways G can act on a vector space. In other words, it's a map from G into a matrix group. Classifying all the representations of a given group is of important interest to many number theorists, and representation theory has many applications outside of math, for instance to quantum mechanics. 

In this DRP project, we will start by reviewing some basic group theory and linear algebra, before delving into the definitions surrounding the representation theory of a finite group. We will look at plenty of examples of representations, and – if you are interested — learn just enough about the mathematical formulation of quantum mechanics to see how representations of certain groups describe how quantum mechanical states change under symmetry transformations (no physics background required)!

Suggested Book: "Representation Theory and Characters of Groups" by James and Liebeck

Prerequisites: Linear algebra and group theory are recommended but not required — the scope and pace of the project can be adapted to your background!

• Proposed project: An Introduction to Group Theory Through a Visual Approach

Description: What do the moves of a Rubik’s Cube, the symmetries of a snowflake, and the encryption behind secure online transactions have in common? They’re governed by the mathematical language of symmetry and group theory. Group theory is a cornerstone of modern mathematics with deep connections to a wide range of disciplines, including computer science, physics, chemistry and cryptography. 

In this reading project, we will focus on developing an intuitive understanding of concepts in abstract algebra such as symmetries, cyclic and permutation groups, subgroups and group homomorphisms. As we progress, we’ll explore how these abstract concepts connect to broader areas of mathematics and real-world applications.

Suggested Book: "Visual Group Theory" by Nathan Carter

Prerequisites: There are no formal prerequisites for this project. The only requirements are curiosity, commitment, and a willingness to learn!

• Proposed project: Is it a Square?... Is it a Circle?... No, It's Homology!

Description: Today, many technological systems across nearly every industry employ machine learning. A popular use of machine learning is classifying shapes. Classifying shapes can be a difficult task; however, topologists and geometers have developed tools to tackle this problem, mainly Topological Data Analysis (TDA). This project will start with an introduction to groups, metric spaces, and data science. Then we will discuss some topics in Topological Data Analysis. Lastly, we will build our own model to classify shapes using tools from TDA to improve upon modern classification models.

Suggested book: "Computational Topology for Data Analysis" by Tamal Dey and Yusu Wang

Prerequisites: Students need to be familiar with Calculus 1, Linear Algebra, and Python or R. Additionally, it would be a plus if students are familiar with Groups and Topology.