# Spring 2023 WXML Projects

Apply until Sunday March 12.

## Lean Learning Lab

**Faculty Mentor:** Dr. Jarod Alper

**Project Description:** Meeting on Wednesdays from 5-6 pm, the Lean Learning Lab (LLL) 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.

Depending on your background and interests, you can choose the mathematical results that you work toward formalizing. 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:**

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

## Mysteries in Metric Spaces

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:** Distance geometry is trying to understand the geometry of a space from understanding the behavior of pairwise distances. Given some general metric space and n arbitrary points, we will be interested in the behavior of the “distance matrix” whose D_{ij} entry is given by d(x_i, x_j). This matrix has a number of interesting properties and we’ll try to investigate them.

There are some interesting universal results and we will try to understand whether we can improve on them when the metric space is not an arbitrary metric space but the metric space induced by a finite graph.

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

**Additional Course Requirements:** 318 (Advanced Linear Algebra) might help but isn’t really necessary

**Programming Requirements:** Very basic Programming skills might help. At least one of the students should be able to do basic program.

## Generalized Matrix Nearness and Homotopy Continuation

**Faculty Mentor:** Dr. Tim Duff

**Project Description:**

We will study, from the perspective of modern computational algebra, a recent preprint of Li and Lim (https://arxiv.org/pdf/2209.14954.pdf) that concerns several different optimization problems generalizing the classical Eckhart-Young Theorem about low rank matrix approximation. A first goal would be be to calculate the Euclidean distance degree of some algebraic formulation of such a problem. A relevant reference for this material is (https://arxiv.org/pdf/1601.07210.pdf). We will also conduct experiments using the numerical computing language Julia (https://julialang.org). In particular, we will learn how to use the software package HomotopyContinuation.jl (https://www.juliahomotopycontinuation.org) to compute the Euclidean distance degree.

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

**Additional Course Requirements:** Substantial exposure to (linear and abstract) algebra, numerical methods, and programming would all be beneficial, but are not necessary for students who are willing to work hard.

**Programming Requirements:** Homework: install Julia, run the following code, and understand what it is doing: https://www.juliahomotopycontinuation.org/examples/computer-vision/

## Quantum Probability via Arbitrary Functions

**Faculty Mentor:** Dr. Ben Feintzeig

**Project Description:** How do deterministic physical systems give rise to probabilities? In classical probability theory, one can show that some differential equations governing the dynamical evolution of physical systems lead to “universal behavior” in the limit of large times. More specifically, the system almost always approaches the same final distribution for certain physical variables, independent of the distribution of initial conditions one began with. This is called the classical method of arbitrary functions—where the arbitrary functions refer to the possible initial conditions, which turn out to be irrelevant to the long time behavior. Such results are thought to explain why physical systems display the probabilities they do, e.g., why a tossed coin lands heads with probability 1/2 or a roulette wheel has equal probability of landing on each number. In this project, we aim to extend the method of arbitrary functions to systems described by quantum physics, where both the dynamical equations and conceptual framework for probability have novel features. We will explore the behavior for concrete physical models in the limit of large times and the classical limit of small values of Planck’s constant.

## Computing dimensions

**Faculty Mentor:** Dr. Silvia Ghinassi

**Project Description:** We will introduce different concepts of dimension, intuitively generalizing what we already know for integer dimensions (e.g., a line is 1-dimensional, a square is 2-dimensional, and so on). These include (but are not limited to) box counting, packing, divider, and Hausdorff dimensions. With these tools at hand, we will compute dimensions of some objects, such as simple fractals or coastlines. My programming knowledge is very limited but the goal of the project is to have the students write programs to compute such dimensions. Depending on time, and student interests and/or mathematical levels we will also talk about fractional measures (but no measure theory is required).

**Project Level:** Open: students who have taken Math 126

**Additional Course Requirements:**

**Programming Requirements:** Since I don’t know any (besides a teeny tiny bit of Python), ideally they’d know something (Python, Matlab are suitable probably ? I don’t know)

## Symmetric Group Actions and Applications

**Faculty Mentor:** Dr. Sara Billey

**Project Description:**

Following up on fall quarter’s special topics course in “Symmetric Group Representation Theory”, we have discussed three open problems and some related literature. Our main focus has been advancing the notion of Graph Modules introduced by Daniel Brosch in his 2022 Ph.D. thesis. Brosch identified these modules as useful in studying problems in Optimization following the work of Raymond-Saunderson-Singh-Thomas and Gatermann-Parrilo. His focus has been on decomposing graph modules into a direct sum of permutation modules. We have been considering the finer decomposition into irreducible S_n modules, connections to matroids, and connections to the enumeration of isomorphism types of graphs.

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

**Additional Course Requirements:** Math 480 fall 2022

**Programming Requirements:**

This project is not accepting new students.