Current Projects

Mentor: Austin Spratt

Area: Basic Commutative Algebra

Overview of the topic: To engage in a slow paced introduction to commutative algebra. This will serve as additional training for undergraduates who wish to pursue algebra later on.

Pre-requisites: Math 3000 (Introduction to Proof Writing). Student must have taken or be currently enrolled in Math 4710 (Introduction to Abstract Algebra).

Text: Commutative Algebra by Altman Klieman

Mentor: Richard Knupp

Area: Mathematical Physics

Overview of the topic: The goal for the semester is to introduce students to mathematical physics while learning a fun application. This will include the rigorous treatment of basic classical and quantum mechanics. As for the application, we will dive into the realm of quantum computing and learn some basic Qiskit to run quantum circuits. Topics intended to be covered, with time permitting, will include the following:

• Classical Mechanics: Newton's Laws, Momentum, Energy, Hamiltonians, Poisson Brackets, and Conservation Laws

• Quantum Mechanics: Hilbert spaces, inner products, Hermitian operators, axioms of quantum mechanics, Dirac notation, and tensor products

• Quantum Computing: Learn about qubits, measurements, gates, and circuits through Qiskit: a programming language based in Python that helps perform quantum operations and measurements.

At the end students will be able to: Demonstrate rigorous treatment of physical ideas and use basic Qiskit to run a quantum circuit

Pre-requisites: MATH 3000 (Introduction to Proof Writing), a linear algebra course such as MATH 4140 or 4920, a calculus course such as MATH 2300 or higher, and some basic understanding of ODE's/PDE's. A little physics and Python experience is good, but not needed. As long as you are willing to learn the necessary physics and Python code, that's all that matters! 

Text: Two books, but I have copies of both! Quantum Theory for Mathematicians, by Brain Hall, and Quantum Computation and Quantum Information, by Nielsen and Chuang

Mentor: Christian Hirni

Area: Algebra

Overview of the topic: The student will explore applications of mathematics (particularly group theory) to describing music theory. Depending on the interests of the applicant(s), the following topics can be covered:

• Neo-Riemannian theory and The Tonnetz,

• Generalized interval systems,

• Symmetries, Invariants, and Representations,

• Applications of Group Theory to Computer Music Generation.

Pre-requisites: Student must have taken MATH 3000 (Introduction to Proof Writing). Nice to have a background in music theory or introduction to abstract algebra, but these not strictly required.

Text: Depending on the interests and prior knowledge of the applicant(s) the following texts may be explored: 

• An Introduction to group Theory- Applications to Mathematical Music Theory, by Lluis-Puebla, et al,

• Generalized Musical Intervals and Transformations by David Lewin,

• Maybe something else, at the request of the applicant(s).

Mentor: Jacob Lawrence

Area: Mathematical Physics

Overview of the topic: One way to study an object is to throw another object at it and track how the object "bounces" off. We will discuss how this works out mathematical and physically. 

Pre-requisites: MATH 4100: Differential Equations is not necessary but is extremely helpful. A background in physics is not necessary, but it is helpful. 

Text: I will use various documents through, but an example is Griffiths' "Introduction to Quantum Mechanics".

Mentor: Kyle Loftus

Area: Analytic Number Theory

Overview of the topic: I’d like to share how we can study arithmetic (How many primes are there? How accurately can we count them? How are they distributed amongst themselves?) through complex-valued functions and tools. Specific topics may (depending on time/pre-requisites/speed/interest) lead toward/include prime number theorem, the Riemann Hypothesis (and generalizations), error terms, Mobius cancellation, and twin primes. 

Pre-requisites: 

Hard requirement: MATH 2300/3000

Can be co-requisites:

Advanced calculus (MATH 4700/4900)

Complex variables (MATH 4940)

…alternatively we can do a smaller amount of number theory and get through most of the relevant parts of prerequisites and/or “accepting” some things as fact, since they themselves are interesting in my opinion 

Text: There will be several sources; I can give ad hoc references when desired. The library has copies of most of them.