# Spring 2024 Projects

## Title Extremal growth in groups: asymptotics and algebraic combinatorics

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

**Project Description:**

The growth of finitely generated groups is a well-studied fundamental invariant in geometric group theory. In this project we will study the ‘extremal growth’ of groups: for each k,n we count the maximum number of length-n words which can be composed of any size-k subset of the group. We will investigate this invariant from both asymptotic and algebraic combinatorial perspectives. For (and only for) virtually abelian groups, if k and n are linearly related (e. g. n=k), the extremal growth is exponential as a function of (each one of) them; we will search for a description of the growth exponent by means of the algebraic structure of the group. For many groups, the extremal growth sequence satisfies linear recurrences (e.g. for abelian groups, this sequence is obtained from binomial coefficients); we will address this phenomenon for virtually abelian groups by identifying the extremal growth sequence with interesting algebraic combinatorial counts; the tools include wreath products, labeled functions and their pattern histograms, formal power series techniques, entropy of random variables and more. Time permitting, we will explore non virtually abelian groups and generalize recent results on the extremal growth of nilpotent groups. The methods we will use include pure mathematical arguments as well as computer simulations to establish and check various enumerative conjectures.

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

**Additional Course Requirements:** Math 403 or knowledge at the level of a first course in Group Theory

**Programming Requirements:** Experience with Sage will be useful (programming language: Python suffices).

## Rotating Cubes in High Dimensions

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:**

We’ll look for ways of rotating the unit cube so that `most’ of its vertices have at least one coordinate that is relatively large (more precisely, what we want is for the average maximal coordinate to be as large as possible). There is also a statistical interpretation: designing fair statistical tests that, when combined, usually end up having at least one test giving an atypical result. The geometric explanation may be easier but visualizing 8-dimensional cubes is also challenging.

The problem is pretty easy in the plane, we rotate the square 45 degrees and that is optimal. It becomes quite a bit trickier in higher dimensions and it seems that there are unexpected solutions with beautiful patterns. We’ll see whether we can find some more good rotations; maybe they have some hidden structure? Maybe something can be proven?

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

**Additional Course Requirements:**

**Programming Requirements:** Basic Programming skills may be useful

## 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:**

## 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

## Cryptography vs Divination Systems

**Faculty Mentor:** Dan Shumow

**Project Description:** THIS PROJECT IS NOT ACCEPTING NEW STUDENTS IN SPRING QUARTER. 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.

## Quantum Probability via Arbitrary Functions

**Faculty Mentor:** Dr. Benjamin Feintzeig

**Project Description:** THIS PROJECT IS NOT ACCEPTING NEW STUDENTS IN SPRING QUARTER. 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:**

## Applications of concentration of measure

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

**Project Description:** THIS PROJECT IS NOT ACCEPTING NEW STUDENTS IN SPRING QUARTER. 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