Fall 2023 Program

Meet the Mentors

Sarah Chehade

Postdoc in Quantum Computational Science at the Oak Ridge National Laboratory

Description: In this project, we will learn about quantum computers, how they’re different from regular, classical computers, and what they’re used for in modern science and technology. We will start with an exploration of how classical computing works and build up to quantum computing. Some topics covered here include representing information with bits and qubits (which is the quantum analogue of bits), as well as performing computations using logic gates and quantum logic gates. To wrap up the project, we will learn about quantum teleportation, the process of sending secret messages through qubits that are entangled.

Suggested Book: "Introduction to Classical and Quantum Computing" by Thomas Wong

Prerequisites: We mostly need a strong background in linear algebra. A course in Probability or Statistics will be very useful. Most of all, we need an excitement to learn and a willingness to accept the fact that “if you think you understand quantum mechanics, you don't understand quantum mechanics.”

Kaya Lakein

PhD Student in Math at the University of Michigan

Description: In this project, we will explore the basic theory of elliptic curves in the (x,y)-plane and study their rational points (i.e. points on the curve whose (x,y)-coordinates are all rational numbers). Elliptic curves are foundational to modern number theory and have applications to cryptography (the art of sending and receiving encrypted messages). They are also very accessible at the undergraduate level!

Over the course of the project, we will encounter an unexpected, but visually meaningful example of a group that you probably did not see in your first course on group theory. We will also gain some practice reading and writing proofs in an environment that blends algebra with geometry. A lot of the readings will be very hands-on, and I will supply recommendations for some related exercises that will allow you to further develop your problem-solving skills. At the end of the semester, you will know how to define an elliptic curve, describe what its rational points look like, and relate them to the curve’s integer points.

Suggested book: Chapters 1-3 of “Rational Points on Elliptic Curves” by Silverman and Tate

Prerequisites: Familiarity with proofs and basic group theory (definition of a group, a subgroup, a group homomorphism, and its kernel) is recommended but not strictly necessary. The pace of the project can be adapted to your background!

Lucas Onisk

Postdoc in Math at Emory University

Description: Solving linear systems of equations is often an integral part of solving much larger problems that arise in computational sciences such as in medical imaging, engineering, and machine learning applications. It turns out that row reduction strategies like Gaussian elimination are often not the most computationally efficient (or stable!) solution methods for every system type. So how do computer programs decide how to solve a given linear system?

In this project, we will explore how the structure of a matrix (think triangular, diagonalizable, block, etc.) helps to inform the fastest and most stable computational solution strategy. Exploring these matrix structures not only introduces material that is often less covered in a typical undergraduate curriculum, but also serves as a natural way to practice approaching problems from a new perspective. Depending on student interest, the focus can be a mixture of coding methods, exploring theory, or implementing methods on real world problems.

Suggested Book: “Fundamentals of Matrix Computations” by Watkins

Prerequisites: A first course in linear algebra. Some coding experience would be helpful, but a little curiosity and a desire to learn are all that is required!

Joshua Stucky

Postdoc in Math at the University of Georgia

Description: Given a set of integers, a natural question to ask is how many divisors a typical member of this set has. For instance, if the set is the set of prime numbers, then all the elements of this set have exactly 2 divisors. What about all the integers up to a thousand, or up to a million? In general, how many divisors does the average integer less than some bound x have? This is a classical question in analytic number theory known as Dirichlet's Divisor Problem. Moreover, the techniques used to answer it are foundational and ubiquitous, and make this question an excellent introduction to the broader field of analytic number theory.

This project sits firmly in the realm of pure mathematics, and will require reading and writing proofs. If the student is not familiar with this yet, we'll spend the first few weeks working through a text on reading and writing proofs. After this, we'll cover a number of techniques and ideas in analytic number theory. Ideally, the project will culminate in understanding Dirichlet's hyperbola method and applying this to give an answer to Dirichlet's Divisor Problem. Time permitting, there are a number of additional topics that could be covered using the tools we will have learned, such as the Gauss Circle Problem.

Suggested book: “Introduction to Analytic Number Theory” by Tom Apostol

Prerequisites: Calculus 1 is required. Calculus 2 and/or a first course in proofs would be nice, but these are not required.

Meet the Students

Jabez Dailey

Nia Legrand

Marley Lucas

Alexandra Pate