Fall 2023 – Mason Experimental Geometry Lab

Fall 2023 Projects are here! Click below for project descriptions. We are accepting applications until Friday, May 12th Click here to apply: https://forms.office.com/r/nypRfTVuQJ

Faculty: Brent Gorbutt

Description:

Our goal will be to find a general formula for the product of two Peterson Schubert classes in non-A Lie types. We’ll use the following papers to guide us:

https://arxiv.org/abs/1311.3014
https://arxiv.org/abs/2004.05959
https://arxiv.org/abs/math/0703637

Faculty: Rachel Kirsch

Description: A De Bruijn cycle is a compact cyclical listing of all the words of length n. For example, the sequence 0011, considered cyclically, is a De Bruijn cycle for n = 2 because the subwords of length 2 beginning in its first, second, third, and fourth positions, respectively, are 00, 01, 11, and 10, which are all of the binary words of length 2. More generally, a universal cycle for a family of combinatorial objects of rank n runs through each element of the family exactly once. Universal cycles have extensive applications in computer science, coding theory, cryptography, and computational biology. This research project examines higher-dimensional analogues of universal cycles such as perfect maps, also known as De Bruijn tori.

This project is either remote or a hybrid of remote and in-person, to be determined by the team.

Prerequisites:

1. Experience with proofs (in courses or research projects) and mathematical maturity of an advanced undergraduate student
2. Experience with combinatorics and/or graph theory (in Math 125, Math 325, or otherwise)

The following experience may be helpful. It is not required.

1. (Optional) Experience with computer programming and/or theoretical computer science
2. (Optional) Experience with abstract algebra, number theory, and/or advanced linear algebra (in Math 301, 321, 322, 421, or otherwise)

Faculty Mentor: Flavia Colonna

Description: In a 1994 paper I wrote with Joel M. Cohen, we solved the problem of how to embed a homogeneous tree of even degree in the unit disk so that the boundary of the tree covers the entire unit circle and such that the edges of the tree are geodesic curves of the same length with respect to the hyperbolic metric. We proved that the explicit construction of such a tree can be carried out using ruler and compass. The purpose of this project is two-fold:

· Provide a visualization of the above geometric construction.

· Using the even degree case as a model, produce a similar construction in the odd degree case, which has not been treated in a published paper.

At the start of the project, we plan to cover the following basic concepts:

· boundary of an infinite tree

· how to view a homogeneous tree as a group

· the hyperbolic disk in the plane

We will then go over the construction described in the paper to obtain a visualization. Then have fun extending this to the odd-degree case.

Recommended Prerequisites: A good grasp of geometry and Math 321. Some exposure to complex analysis in one variable (Math 411) is desirable but not required.

Faculty: David Walnut

Description: The aim of this project is to explore a conjecture about the so-called Fourier matrix (also called the DFT matrix). The conjecture arose in the context of some recent research and if settled positively would result in at least one publication. Given $N$, the $N \times N$ matrix DFT(N) consists of powers of the $Nth$ complex roots of unity, specifically the $(i,j)$ entry of DFT(N) is $\omega^{jk}$ where $\omega = exp(2 \pi i/N)$ and $i^{2} =-1$. DFT(N) is an orthogonal, symmetric matrix with Vandermonde structure. A principal minor of any square matrix is the determinant of the square submatrix formed by fixing a set of columns and choosing entries from the same set of rows. Conjecture: For every $N$, there exists a permutation of the rows of DFT(N) such that every principal minor of the permuted matrix is nonzero. It is known that the conjecture is true whenever $N$ is prime (Chebotarev’s Theorem). Some informal numerical work has been done on this conjecture and it has been confirmed up to $N = 20$ by randomly searching through the space of row permutations. Exhaustive enumeration of principal minors for all permutations has been done for small $N$. The search space grows very rapidly with $N$ so there are challenges to investigating numerically.

Here are some initial steps in the project. 1.Reproduce what has already been determined numerically using MATLAB or other software. 2.Using faster machines, more time, or optimizing code, can we confirm past $N=20$? 3. Special cases like $N=2K$, $N=p^{2}$ where $p$ is prime, or $N=pq$ where $p$ and $q$ are prime can be considered more carefully and thoroughly. 4. Is there a way to display or visualize results for larger values of $N$?

Recommended Preparation: Students working on this project should be comfortable with programming at least in MATLAB. At the start we can just do some numerical investigation, but we will be talking about some other topics like the Discrete Fourier Transform, theory of determinants, and maybe some elementary number theory or group theory. Students will need Math 203 or equivalent. Ideally, students would also have taken Math 301 or Math 321, and be willing to study some topics

Faculty: Michael Jarret

Description: An essential tool in understanding the power of quantum computing is understanding its best classical competition. This can lead to insights into where the power of quantum computation comes from, as well as novel classical and quantum algorithms. In this project, students will analytically investigate numerical algorithms presently used for understanding quantum systems, in order to understand their running time and complexity. Students may be asked to develop novel classical/quantum algorithms, either provable or heuristic.

Faculty: Douglas Eckley

Project Description:

Background

This project will apply the Kelly Investment Criterion to various real-world investing and/or gambling scenarios.

The goal of the project is to bring together several concepts from statistics and economics in a way that that is conducive to visualization. The project can be done entirely either in spreadsheet or in R.

Assignment

1 Research the Kelly Investment Criterion via a textbook that I will supply

2 Present the development of the Criterion

3 Apply the Criterion in various settings, including a setting that leads to the Proebsting Paradox

4 Present results

The statistical concepts involved are:

Utility theory

Maximizing a function

Ruin theory

Call options on common stock

Black-Scholes option pricing

The prerequisites are: 1) derivative calculus; 2) some exposure to continuous probability distributions; 3) some familiarity with spreadsheets or the R statistical package.

No familiarity with spreadsheets or R could be addressed by two weeks of preparatory self-study.

Faculty Mentors: Evelyn Sander and Dan Anderson

Description: Our goal in this project is to use a combination of analysis, computation, and experiments with 3D printed floating objects in order to better understand the method in which biology solves the problem of locomotion via the use of surface tension.