Winter 2021 projects

Random Inelastic Billiards

Faculty Mentor: Dr. Jayadev Athreya

Project Description: We’ll illustrate the dynamics of sticky particles on the real line, using a mixture of probability theory, dynamical systems, programming, and illustration. Our work is motivated by the beautiful work of Ryan Hynd.

Team Members:
Project Level: This project is open to all students who have taken Math 126.
Additional Course Requirements: Ideally students will have taken Math 324 and some intro physics.
Programming Requirements: Preferably know some basic Python or Javascript.

Constructing Quantum Theories

Faculty Mentor: Dr. Benjamin Feintzeig

Project Description: This project investigates the properties of so-called deformation quantizations that model the relationship between classical and quantum physics. We will analyze various differential equations representing distributions of matter in space. Deformation quantization considers algebras of complex-valued functions on the space of solutions to such differential equations, and defines a non-commutative product capturing the fundamental features of quantum physics. Since the deformed non-commutative product approximates a classical commutative algebra in a precise sense, one can use this structure to investigate the “classical limit” of aspects of quantum physics. We will be especially concerned with analyzing the classical limit of quantities associated with energy and particle content in physics.

Team Members:
Project Level: This is an advanced project.
Additional Course Requirements: Some background in linear algebra, analysis, and abstract algebra will be helpful for this project. No background in physics is required.
Programming Requirements:

Geometry of zeros of complex polynomials

Faculty Mentor: Dr. Harry Richman

Project Description: The goal of this project is to explore patterns between the location of zeros of a polynomial in the complex plane and the zeros of its derivative. More specifically, we will look for ways to generalize a basic observation known as Rolle’s theorem: for a real-valued polynomial p(x), between any two (consecutive) zeros of p(x) there must be a zero of p'(x). The case of complex polynomials is much less well-understood. For complex polynomials, the relation between zeros of p(x) and zeros of p'(x) is equivalent to the following electrostatic problem: given the position of n electrons fixed in the plane, where could I place an additional electron such that it would not be pushed around by the fixed electrons?

We will explore this problem using computational visualization tools, using Mathematica or python. (See here for example.) Using these observations on specific families of polynomials, we will then formulate conjectures about what happens for general complex polynomials and try to prove them.

Team Members:
Project Level: Open: students who have taken Math 126
Additional Course Requirements: in addition to calculus, familiarity with differential equations or electromagnetism is preferred
Programming Requirements: some experience with programming; willingness to learn Mathematica and/or python

Random Walk Excursions and the Stochastic Sandpile Model

Faculty Mentor: Dr. Christopher Hoffman

Project Description: We will look at random walk excursions on Z . This is the path of a random walk that starts at the origin until it returns to the origin for the first time. We count the number of times each integer is visited. This is known as the occupation time. We will study the occupation time and see how this relates to a model called the stochastic sandpile model.

Team Members:

Project Level: Advanced: students who have taken multiple upper-level mathematics courses
Additional Course Requirements: math 394/5/6 preferred
Programming Requirements: Proficiency at coding is a plus