# Autumn 2024 Projects

## A strange dynamical system in the plane

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:** We will investigate a new type of sequence of points in the plane that exhibit very intricate behavior (that is also very beautiful to look at). It seems that they always end up in a union of concentric circles and it’s not so clear why that is.

This is naturally related to some recent work of Bettin-Molteni-Sanna about greedy harmonic series and there might be some interesting variations worth exploring. Let’s make some progress.

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:**

**Programming Requirements:** Basic programming skills could be useful.

## Stability of capillary surfaces

**Faculty Mentor:** Dr. Jonathan Zhu

**Project Description:** Capillary surfaces model the equilibrium state of interfaces between fluids. The goal of this project is to determine whether some known examples of capillary surfaces are stable equilibria, using numerical methods or otherwise.

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** required: multivariable calculus, differential equations. may be helpful: numerical analysis

**Programming Requirements:** some experience with MATLAB/Mathematica/Maple/etc

## Nilpotency and geometric structures in noncommutative algebras over finite fields

**Faculty Mentor:** Dr. Be’eri Greenfeld

**Project Description:** In classical algebraic geometry, coordinate rings are commutative and free of nilpotent elements. This project explores noncommutative generalizations of this phenomenon in the context of noncommutative algebra. It is known that, over sufficiently large fields, algebras with an abundance of nilpotent elements do not admit point modules, which are the noncommutative analogs of points in classical geometry. Our objective is to quantify this phenomenon by establishing optimal bounds on the size of the ground field in relation to the nilpotency indices of elements that still allow for the existence of point modules. The work will involve a combination of ring theory and combinatorics, particularly focusing on projective spaces over finite fields. The results we aim to prove will refine and extend known results in the existing literature.

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** MATH 402

**Programming Requirements:** Not necessary. Basic Python skills could be useful.

## Finite Element Methods for Advection-Diffusion Equations

**Faculty Mentor:** Dr. Heather Wilber

**Project Description:** The advection-diffusion equation is a fundamental partial differential equation that models the transport and dispersion of substances within a physical system, such as pollutants in air or water. Through Finite Element Methods (FEM) we will approximate the solution to the advection-diffusion equation. We will begin by studying the mathematical formulation of the advection-diffusion equation, followed by an analysis of different FEM techniques, including standard Galerkin methods.

The project will culminate in applying these methods to real-world problems, such as modeling pollutant transport in environmental systems or heat distribution in engineering materials, with an emphasis on verifying the numerical results against analytical solutions or experimental data.

**Project Level:** Advanced: students who have taken multiple upper-level mathematics courses

**Additional Course Requirements:** MATH 224 (required), MATH464 (useful)

**Programming Requirements:** Julia, MATLAB (should be comfortable with coding)

## Implementing Undetectable Backdoors in Machine Learning Models

**Faculty Mentor:** Daniel Shumow

**Project Description:** A recent paper showed how to implement undetectable backdoors in machine learning models. href=”https://ieeexplore.ieee.org/document/9996741″

The work is fairly theoretical, and it is unclear how difficult this would be to implement in practice. As such, the purpose of this project will be to go over this paper, understand it, and then try to implement these backdoors.

**Project Level:** Intermediate: students who have taken Math 300

**Additional Course Requirements:** Linear algebra would be helpful.

**Programming Requirements:** Python

## Experimental Lean Lab (XLL)

**Faculty Mentor:** Vasily Ilin, Dr. Jarod Alper, et. al.

**Project Description:** Meeting on Mondays from 11:30-1 pm, the Experiemntal Learning Lab (XLL) is dedicated to formalizing mathematics with the Lean Theorem Prover. Lean is a programming language that allows you to verify the proofs of mathematical statements. At the same time, Lean provides high-level tactics that can also assist you in writing proofs by having the computer itself fill in annoying technical details and computations. Theorem provers such as Lean are changing how mathematical research is done so you might as well get on board now!

The main goal of this project is to learn the functionality of Lean by formalizing various exercises and theorems in undergraduate mathematics. As we get better at Lean, we may endeavor to formalize interesting foundational results (e.g. Hilbert’s 1890 proof of the finite generation of invariant rings) and perhaps even contribute to the growing mathematical library of Lean.

To see whether this might be the right project for you and to get started learning Lean, you can play the “Natural Number Game” (https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/).

**Project Level:** Intermediate: students who have taken Math 300

**Additional Course Requirements:** at least one additional 300 proof based level course

**Programming Requirements:** Experience with programming is helpful but not required. We will learn Lean programming language together.