# Autumn 2023 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:**

## Pinned billiard balls

**Faculty Mentor:** Dr. Krzysztof Burdzy

**Project Description:** Pinned billiard balls is a model of billiard balls spaced so tightly that they can assumed to be stationary but have non-vanishing (pseudo-) velocities. I would expect students to write simulations of the evolution of velocities and verify some hypothesis about the evolution and terminal distribution of energy. If students generate and test their own conjectures, that would be great.

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

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

**Programming Requirements:** Knowledge of at least one language that can be used for simulations.

## 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:** Dr. Jarod Alper, 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.

## The Structure of Balanced Measures

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:** Balanced measures on graphs are probability distributions with a particularly nice extremal property: moving all the probability weight into a single vertex is maximal in exactly the vertices where the measure is supported. We will study ways of finding balanced measures, hope to determine the rate at which the greedy construction method converges and will try to understand their structure a little bit better.

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

**Additional Course Requirements:** 407/408 (Optimization) might help but is not required.

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

## Computing Dimension

**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:** Basic programming skills might be useful.