# Winter 2024 Projects

## Quantum Probability via Arbitrary Functions

**Faculty Mentor:** Dr. Benjamin 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.

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

**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:**

## Cryptography vs Divination Systems

**Faculty Mentor:** Dan Shumow

**Project Description:** Divination systems, such as Astrology or Tarot, have been used though out human history, in cultures across the whole world. Though they are based in outdated understanding of the universe, rooted in religion and superstition they have remained popular. Astrology is based on the positions of the sun, moons and other planets in our solar system relative to constellations (distant stars), Tarot is like a deck of cards, and the divination technique follows a regular pattern of drawing cards. This can be viewed as analogous to cryptographic pseudorandom number generators, which provide randomness to cryptographic algorithms and protocols, and how they extract randomness from the environment

This project is interested in looking past the unscientific origins of these systems, and instead seek to characterize the system as a deterministic system that takes readings from the environment. Then these algorithms will be analyzed to see if they provide the desired security properties that a cryptographic random number generator does.

This project also provides an opportunity to make graphical representations of the systems involved, such as the mapping of the actual solar system onto the astrological system.

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

**Additional Course Requirements:**

**Programming Requirements:** Basic programming skills such as Python or using Sage would be good, but not required if the student has Math 300 or above.

## Experimental Lean Lab (XLL)

**Faculty Mentor:** Drs. Andy Heald, James Morrow

**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.

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.

## Intrinsic Dimension in Data Sets

**Faculty Mentor:** Dr. Silvia Ghinassi

**Project Description:** Last quarter, our team defined a notion of dimension for finite point sets, analogous to familiar notions in Geometric Measure Theory of dimension for “continuous” sets. We computed our intrinsic dimension for a few familiar sets of dimension between 0 and 2, and started testing on some data sets. This quarter we plan on generalizing our definition of “intrinsic dimension” to higher dimensions, and to continue testing our notion in data sets that arise from the real world. We hope that interdisciplinary collaborations will help us interpret our findings.

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

**Additional Course Requirements:** Analysis Courses (Math 327, 424, etc.) are prefered

**Programming Requirements:**

## Title Combinatorial, computational and homologcial aspects of quadratic algebras

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

**Project Description:**

The aim of this project is to use computational tools to obtain a conjectured optimal bound on the nilpotency degree of finite-dimensional quadratic algebras with a fixed number of generators, and ideally, prove it. This would have interesting combinatorial and homological applications: for instance, it will give an effective bound on the vanishing of the Koszul complex of quadratic algebras (if finitely supported), and thereby on their global dimensions.

The problem is combinatorial in nature and can be understood by means of linear algebra and combinatorics of words; (basic) ring theory would be helpful. The first approach we will take is geometric, by calculating the parametrizing varieties of these algebras. The second approach will concentrate on semigroup algebras, which are more restricted than the general class, and the problem can be checked for them by smartly going over all of them (for small numbers of generators). Finally, we hope to tackle the full class of finite-dimensional quadratic algebras.

Despite the deep roots of the theory, I expect the project to be accessible to students with computational skills and with a reasonable familiarity with rings (especially polynomial rings) — making it a great opportunity for undergraduate students to experience a research project with interesting potential results.

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

**Additional Course Requirements:** MATH 402 would be helpful (but not necessary)

**Programming Requirements:** Familiarity with Sage would be useful (but it is easy to work with it once you can write in Python)

## Wave propagation on graphs

**Faculty Mentor:** Dr. Hadrian Quan

**Project Description:** In addition to describing real networks and datasets, graphs also provide discrete models of continuous geometric objects. On a smooth surface the behavior of solutions to the wave equation is strongly influenced (and describes) the geometry of the underlying surface; how do such phenomena arise in graphs, and can the behavior of waves be used to understand properties of complex graphs? This project will explore such questions, and seek to apply these insights to questions such as link prediction in networks.

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

**Additional Course Requirements:** Math 207/208/318 and any courses with exposure to graph theory are preferred but not required

**Programming Requirements:** some experience with Python/mathematica is preferred but definitely not required

## Title Mathematics of Gerrymandering

**Faculty Mentor:** Dr. Christopher Hoffman

**Project Description:** Every decade each state is divided up into

districts to elect members of Congress. Modern mathematical tools have allowed the people who draw the maps to gain significant political advantages. Other mathematical tools allow us to quantify how much a given map favors the two parties. In this project we will learn about all of these mathematical tools (including Markov Chain Monte Carlo and generative AI) and analyze the current map for the state of Washington.

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

**Additional Course Requirements:** Probability courses (Math 394/5/6 or equivalent) are a plus but not required.

**Programming Requirements:** Programming experience is a plus

## Applications of concentration of measure

**Faculty Mentor:** Drs. Hadrian Quan and Andrea Ottolini

**Project Description:** Suppose we have many random variables $X_1,\ldots, X_N$, how large can we expect their sum to be? If they all “cooperate”, you can expect this sum to be of size $\sim N$, but if we assume that they are independent and of bounded range, this sum typically concentrates in a much smaller range of size $\sqrt{N}$.

This phenomenon occurs much more generally, and it is referred to as concentration of measures: a small (in a metric sense) region contains a large (in a probabilistic sense) mass. The goal of this project is to investigate how to generalize this idea and understand the geometric and probabilistic ideas behind it.

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

**Additional Course Requirements:** Math 394 is useful but not required.

**Programming Requirements:** Some programming experience is useful but not required