# Autumn 2022 Projects

## Permutation polynomials

**Faculty Mentor:** Dr. Masahiro Nakahara

**Project Description:** A permutation polynomial over a finite field is one that permutes the elements of the finite field. For example, over F_3, the polynomial x^3+x is a permutation polynomial. It sends 0 to 0, 1 to 2, and 2 to 1. It can be shown that there exists polynomials that give rise to every permutation possible on any finite field F_q. However, it is often a hard question to write down permutation polynomials of low degree and also to test any given one.

The goal of this project is to explore what we can say about these polynomials and get hands-on experience with them. Some questions to consider are what kind of permutations can be obtained once we fix a particular property (such as degree)? A polynomial can be thought of as a function from the F_q-line to itself. What about functions from the F_q-plane to itself?

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

**Additional Course Requirements:** Some familiarity with finite fields would be ideal.

**Programming Requirements:** Not required but could be helpful

## 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.

## Extremal closed geodesics

**Faculty Mentor:** Dr. Stefan Steinerberger

**Project Description:** We will work with a completely explicit function (assuming only values plus/minus 1) on the 3-dimensional flat torus (so it’s really the usual unit cube in 3 dimensions with boundaries glued together like in a video game, you exit on the left means you re-enter on the right and so on). We will consider the set of all closed geodesics and try to identify geodesics on which the function behaves `most atypically’.

The motivation for this comes from quantum mechanics. One expects that particles in a coffee mug care what nearby particles are doing and that they care slightly less what currently happens on the moon. This seems to be mainly true in d=3 dimensions but fails in a somewhat subtle way for d=1. So the project could also be described as trying to understand long-distance correlations of eigenfunctions of Schrodinger operators assuming a WKB asymptotic.

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

**Additional Course Requirements:**

**Programming Requirements:** some programming may help, the language is not important

## Homotopical combinatorics

**Faculty Mentor:** Dr. Kyle Ormsby

**Project Description:** Homotopical combinatorics lies at the intersection of abstract homotopy theory, category theory, and enumerative combinatorics. Its fundamental goal is to uncover enumerative and structural properties of weak factorization systems, premodel structures, and model structures on complete lattices (partially ordered sets admitting all meets and joins). This is a new field with a lot of open questions, often reducible to elementary (but tricky!) combinatorial problems; solving these problems pays immediate dividends in homotopy theory. Students will work collaboratively to learn the fundamentals of the subject and explore open problems via enumerative, analytic, and computational methods. For a taste of the subject, see https://kyleormsby.github.io/posts/2021/09/homotopical-combinatorics/.

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

**Additional Course Requirements:** strong grounding in elementary combinatorics required; familiarity with group theory and categories could be beneficial

**Programming Requirements:** familiarity with Sage or other computer algebra systems could be beneficial

## Hypergraphs from Network Data for Cybersecurity

**Faculty Mentor:** Drs. Bill Kay and Stephen Young

**Project Description:** Network data is typically a collection of log files that contain information about node-to-node interactions (timestamps, IP address, ports, etc.). These files are large and complex. Correlations between log entries varying with time are of interest to cybersecurity analysts. Namely, if we know what normal network activity looks like, can we deduce if some anomalous (potentially malicious) activity occurs?

One way to represent this data is with a hypergraph. Hypergraphs are generalizations of graphs which represent multi-way relations (like peer groups or author lists) rather than just pairwise relations. Properties of a hypergraph which models real life systems can often provide insight into complex relationships that are difficult to deduce from raw data. In this project, students will use an open cybersecurity dataset (https://github.com/FiveDirections/OpTC-data/blob/master/README.md) which has both normal and anomalous network activity to experiment various hypergraph models. They will use tools from hypergraph visualization (HypernetX) and scientific computing (Network X, Neo4j, numpy, etc.) to analyze which hypergraph features of each model can be used to signal anomalous network activity.

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

**Additional Course Requirements:** Discrete Mathematics

**Programming Requirements:** Python

## Analyzing the ‘shape’ of data – persistent homology

**Faculty Mentor:** Dr. Zhixu Su

**Project Description:** Topology studies features of geometric objects that are preserved under continuous deformation. Topological data analysis uses techniques from topology to understand the ‘shape’ of data set, which can be high dimensional and noisy. Given a point cloud data set, one can construct a nested sequence of complexes thickening the data set at different scales. Persistent homology detects topological features of the complexes that persist over a range of scales, it computes the numbers of connected components, 1-dimensional holes (loops), 2-dimensional holes (voids), etc, and keeps track of the birth and death scale of the features. A long lived hole is deemed as significant topological feature, while a short lived one may be interpreted as noise. Persistent homology is computable by linear algebra and there are multiple open-source software implementations for the computation.

In this project, we will explore various methods of building filtered simplicial complexes and understand the standard algorithm for computing persistent homology. Using existing software packages, we will work through exercises computing persistent homology of synthetic data.

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

**Additional Course Requirements:**

**Programming Requirements:** Experiences with Matlab or Python will be helpful.