• Proposed project: The Mathematics behind Quantum Computing

Description: Quantum computing is a branch of computer science that has been in the news lately. Due to special properties of quantum states, quantum computing offers strong computational power. For example, classical computers cannot factorize an arbitrarily large integer into primes within polynomial time; however, quantum computers can do this with Shor’s algorithm. One standard encryption method, called RSA encryption, is based on factorizing large prime numbers. If a strong enough quantum computer could be built in the future, then the RSA encryption would no longer be secure.

Even though quantum computing offers high computational power, it is currently hard to conduct many classical computational tasks in quantum computers. It is difficult to build a good quantum computer, and designing quantum algorithms is hard and different from classical computing. Therefore, more research is needed in this field.

In this project, we will explore the basic mathematical principles behind quantum computing and our journey will follow the book "Introduction to Classical and Quantum Computing" by Thomas Wong. While the subject may appear complex, we will take a step-by-step approach, beginning with a review of classical computing and then moving onto basic quantum computing concepts like quantum bits (aka qubits), quantum gates, and quantum algorithms. Depending on the student’s interest, we can then delve into more advanced topics like the quantum Fourier transform, quantum error correction codes, or actual implementations of quantum algorithms on real quantum computers.

Prerequisites: We mostly need a strong background in linear algebra. A course in Probability or Statistics will be very useful.

• Proposed project: Squaring back to the 1700s

Description: Number theory has a reputation for having problems that are easy to state, but whose solutions rely on very technical methods. In this project, we will study some of the more elementary aspects of number theory. Our first goal is to survey the first part of the book; these form the main foundation of elementary number theory. From here, the student can pick what intrigues them the most from the remaining chapters and we can work our way up to that. We will do problems together every week, and hopefully by the end of the semester we will be able to understand Lagrange's four square theorem (which says that every natural number can be written as the sum of four squares), as well as the original proof of the theorem, given by Lagrange in 1770.

Suggested Book: "A Friendly Introduction to Number Theory" by Joseph H. Silverman

Prerequisites: A proof-based writing course would be a really helpful background for this project.

• Proposed project: Lights Out Type Games

Description: "Lights Out!" is an electronic game from the 1990s consisting of a 5x5 grid, each square of which is either ON or OFF. At the start of the game, a random configuration of squares is turned ON. By pressing a square, the user turns it and all adjacent squares from their current state to their opposite state. The objective of the game is to turn all squares OFF.

Several questions arise from the nature of the game. Can the user win, regardless of the starting configuration? Is there an optimal strategy? Solutions to "Lights Out!" can be described mathematically, primarily via Linear Algebra. Beyond the classic version of "Lights Out!," there are a handful of variants, each of their own mathematical interest.

This project lives in the realm of "recreational mathematics." Recreational mathematics are an excellent way to get a taste of research in mathematics and see where all those techniques you learned in class can actually be used.

Suggested Book: Most material is free online, but a paper that has been useful in the past is "Turning Lights Out with Linear Algebra" by Anderson and Feil

Prerequisites: A first course in Linear Algebra is needed. A course in proof-writing would be good, but is not necessary.

• Proposed project: Introduction to Cryptography

Description: This project is a broad survey of mathematical cryptography. We will start in Chapter 1 with ciphers and the basics of number theory and algebra required for this book. We will then spend most of the semester on public key cryptosystems and study several real-world examples – Diffie-Hellman, ElGamal, and RSA. We will learn how to encode/decode each of these systems by hand and discuss the computational problems that determine how “hard” these systems are to crack. If time permits, we will then move onto more complex cryptosystems using elliptic curves and/or lattices.

Suggested Book: "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher, and Silverman

Prerequisites: Only linear algebra is required, but we can move faster if the student knows number theory and/or abstract algebra.

• Proposed project: Knot Theory and Low-Dimensional Topology

Description: If you know how to tie your shoes, then you’re more than ready to become further acquainted with the branch of mathematics known as knot theory! While this discipline has applications to biology, chemistry and physics, it’s an interesting and thriving mathematical field in its own right. Our project will begin with an introduction to knots, links and braids, and will focus on the problem of determining when two of these objects are the same. We’ll continue with an exploration of other topics in low-dimensional topology. This project is best suited for students with an interest in geometry and topology.

Suggested Book: "An Interactive Introduction to Knot Theory" by Allison K. Henrich and Inga Johnson

Prerequisites: Calculus 3 and Linear Algebra. A course on proofs would be a plus.

• College: Spelman

• Class year: '25

• Major(s): Economics, Mathematics Minor

• Working with: Henry Potts-Rubin

• College: Morehouse

• Class year: '25

• Major(s): Mathematics

• Working with: Shilpi Mandal

• College: Spelman

• Class year: '25

• Major(s): Mathematics, Engineering

• Working with: Robin Wilson and Alissa Crans

• College: Spelman

• Class year: '26

• Major(s): Mathematics, Engineering

• Working with: Cordelia Li

• College: Spelman

• Class year: '25

• Major(s): Mathematics, Pre-Law

• Working with: Juliet Whidden