# NUMS 2023 Abstract list

Abstracts are arranged below by topic and then alphabetical. A link to the schedule of talks will be posted here soon.

### Combinatorics

- In a 2016 paper, Straub proved an analogue to Euler's celebrated partition identity for partitions with a fixed perimeter. Later, Fu and Tang provided both a refinement and generalization of Straub's analogue to $d$-distinct partitions. They also prove a related result to the first Rogers-Ramanujan identity by defining two new functions, $h_d(n)$ and $f_d(n)$ for a fixed perimeter $n$, that resemble the preexisitng $q_d$ and $Q_d$ functions. Motivated by generalizations of Alder's ex-conjecture, we further generalize the work done by Fu and Tang by introducing a new parameter $a$, similar to the work of Kang and Park. We observe the prevalence of binomial coefficients in our study of fixed perimeter partitions and use this to develop a more direct analogue to $Q_d$. Using combinatorial techniques, we find Alder-type partition inequalities in a fixed perimeter setting, specifically a reverse Alder-type inequality.
- Given a finite, complete graph $G=(V,E)$ with some probability measures supported on it's vertices, we can define a subset of these probability measures that have nice properties. We call such measures \textit{balanced}. The structure of such measures has some interesting properties that we're currently exploring. One such property is that, given the graph join of $n$ graphs, we're able to define a unique balanced measure for each subset of the graph join. Furthermore, given a set of graphs $G_i$ that each meet certain conditions, we've been able to find an upper bound on the number of balanced measures that can be supported on the graph join of $n$ such graphs.
- What does it mean for graphs to have boundaries? Recall that the boundary of subsets in $\mathbb{R}^n$ is induced by a metric. Graphs also have a metric, and thus one may try to find a subset of vertices in a graph which may behave like a "boundary" to the graph. Steinerberger proposed a notion of a boundary for graphs which also establishes a corresponding isoperimetric inequality: larger graphs should have more boundary vertices. For our research, we characterized all graphs with two, three, or four boundary vertices. We characterize all graphs with three boundary vertices as belonging to one of two infinite families of graphs. We also characterize all graphs with four boundary vertices as belonging to one of eight families of graphs, five of the families being infinite. Our work parallels previous work done by Hasegawa and Saito, as well as Müller, Pór, and Sereni for a different notion of graph boundaries defined by Chartrand, Erwin, Johns, and Zhang. This project was done under WXML, a math research lab at the University of Washington, under Dr. Steinerberger and Kevin Liu.
- Take a finite, simple, connected graph $G = (V,E)$. Consider it's distance matrix $D$, whose $(i,j)$th entry is $d(v_i,v_j)$. In previous work by Steinerberger (J. Graph Theory, 2023), it was empirically seen that the all-ones vector is frequently lies in the image of $D$, even when $D$ is not invertible. We present some findings towards understanding this phenomenon, including counterexamples for graphs of all sizes $n \geq 7$ and a proof that $D$ is invertible with high probability.
- Polyomino Visibility Graphs (PVGs) have a 2-dimension or 3-dimension polyomino as each vertex, and they have an edge if there is a non-degenerative rectangle or rectangle prism between them respectively. In this talk, we will show that all PVGs can be represented in 2.5 dimensions, 3D but all polyominoes must touch the xy-plane. We utilize a depth-first algorithm to create a polyomino ordering that can be connected with two separate constructions; handles and nubs.

### Number Theory

- K-tuples are sets of prime numbers. Further, prime constellations are k-tuples

that have primes as close together as possible. Prime numbers are positive

integers that have no factor except 1 and themselves. There are no negative

prime numbers. The first few prime numbers are 2, 3, and 5. The number 101

is prime. But the number 1001 is composite. The factors of 1001 are 7, 11, and 13.

There are infinitely many prime numbers. So, there is no largest prime number.

However, as of October, 2023, the largest prime number known to human kind is

the 51st known Mersenne prime number. Mersenne prime numbers have the form

2^n-1. The largest one currently known is 2^(282,589,933)-1. That number has

24,862,048 digits. So more than 24 million digits. The Great Internet Mersenne

Prime Search (GIMPS) project has been going since 1996. You can also download

software and join the search for a larger prime number. Do it for the sport of it.

You could be a prime number hunter. - A widely known and thought about open question in number theory is whether or not an odd perfect number, that is an odd positive integer equal to the sum of its proper divisors, exists. In 1888, James Joseph Sylvester proved that such a number cannot be divisible by 105. This presentation includes an extension of Sylvester's proof discovered recently by the speaker and lists more integers that can be proven not to divide an odd perfect number. We will also discuss how to find such integers.
- An integer lattice point is said to be visible if a direct line of sight from the origin can be drawn to it. Visible and invisible lattice points can be classified by the greatest common divisor (gcd) of their components. If we instead consider curved lines of sight, the standard gcd is no longer sufficient to classify lattice point visibility. In this talk we discuss properties of a generalized gcd called the weighted greatest common divisor (wgcd) and how it can be used to classify lattice point visibility along parameterized curves in higher dimensions.

### Mathematical Aspects of Computer Science

- Visually representing multidimensional vectors in spaces above $\mathbb{R}^{3}$ for analyzing correlations and statistical traits remains a significant challenge, particularly within machine learning applications such as clustering analysis. While Inselberg's Parallel Coordinates (1997) set a foundation, the CWU Visual Knowledge Discovery and Imaging Lab expanded on this with Generalized Line Coordinates (Kovalerchuk, 2018). Building on this, we present a system starting from Static Circular Coordinates plotting connected points along a circumference sectioned by n-D vector components $x_1, x_2, \ldots, x_n$. Instead of static sectioning of the circumference, vectors can be plotted as subsequent sections building Dynamic Circular Coordinates. Then, Linear Discriminant Analysis optimizes the coefficients of plotted linear functions. Lastly, computer-aided visually navigated multidimensional scaling can further tune the model. This system allows for analytical prediction of previously unseen vectors using rule-based classification chaining determined from previous vector data from the same domain. By including proportional axes as needed both dual and multi-class comparisons can be made.
- Mental health issues among college students have become a prevalent concern in recent years. Understanding the main struggles and identifying factors that correlate with poor mental health can provide valuable insights for institutions to better support their students. In this project, we aimed to analyze mental health issues among college students using data from the Reddit platform. By leveraging the Reddit API and applying various natural language processing techniques, we sought to gain insights into the emotional well-being and challenges faced by college students. We will explain how we perform topic modeling with BERT and share our insights.
- Proof assistants such as Lean have taken on an increasing role in Mathematics, even now being used to verify the correctness of difficult theorems in modern mathematics. This talk aims to introduce the Lean proof assistant, its applications to mathematics and other fields, and the speaker's personal experience with it.

### Mathematics Applications in the Sciences

- Developing efficient and accurate methods to model the Hall Effect Thruster (HET), a thruster often used in satellite propulsion, is of great interest to an aerospace research organization. Conventionally, models of the HET are necessarily extremely computationally expensive (“high fidelity” models) to achieve acceptable accuracy. One such high fidelity model applies Direct Simulation Monte Carlo (DSMC) methods. This project combined less costly low and medium fidelity models into a single multifidelity model that produced a solution with comparable accuracy to DSMC in significantly less run time. Considering the simplified scenario of supersonic air experiencing a shock as it travels over a wedge, this model predicts the density of this air across space and time when varying initial densities and velocities. The low fidelity approach uses Proper Orthogonal Decomposition (POD) in a data-fit model based on limited DSMC simulation data. The medium fidelity approach implements a simplified physics based model using concepts from Computational Fluid Dynamics (CFD) solving a set of PDEs. By combining POD and CFD models into a multifidelity framework using Space Mapping techniques, we produced a model with 87% accuracy that runs orders of magnitude faster than the high fidelity model. This novel model drastically reduces computational cost while maintaining accuracy, making it a valuable tool for use in predicting thruster behavior.
- The condition PCOS (Polycystic Ovarian Syndrome) is an affliction that affects approximately $10\%$ of those who menstruate. The defining characteristics of PCOS are hyperandrogenism, irregular menstruation, and polycystic morphology in the ovaries caused by hormonal imbalances during menstruation. Since there is a focus on hyperandrogenism as a symptom, the research modeled the hormonal imbalances with a focus upon testosterone, the prevalent androgen during menstruation, and the introduction of anti-androgens. The modeling of PCOS hormones in an attempt to find an effective treatment is a relatively new endeavor, thus modeling these hormonal imbalances to find ways to lessen symptoms is imperative. Previously, research towards PCOS treatment has mainly focused upon insulin treatments, which have been shown to improve symptoms, and attempting to model the effects of an anti-androgen treatment has been less studied. In order to model these hormonal levels, we modified the differential equations, that Erica Graham put forth in her paper, to focus on pre-menstrual testosterone dependent follicle growth, in an attempt to better model testosterone levels during the menstrual cycle. A bifurcation analysis was unable to be performed on our model due to system and time limitations, which resulted in the model being unstable. Once a proper bifurcation analysis has been done, and if it produces a stable result, one can model the hormonal affects of hyperinsulinemia or of PCOS.

### Mathematical Physics

- Systems of pinned billiard balls serve as simplified models of collisions, where all particles remain fixed in their positions while their (pseudo-)velocities evolve in accordance with the laws of conservation of energy and momentum. Under free-boundary conditions, Athreya, Burdzy, and Duarte have established the maximum upper bound for the number of pseudo-collisions, thereby demonstrating that the number of collisions is finite. In this project, we do extensive numerical simulations to study two alternative environments. First, we consider the balls arranged in half-space with a single ball with inward (pseudo-)velocity. Numerical simulations suggest that in equilibrium, most of the energy is concentrated near the boundary. Second, when the balls are arranged on a flat torus, we find that in the stationary regime, the distributions of the velocity components are i.i.d. normal, and therefore the energies of the balls are exponentially distributed. Additionally, we find that the components of the velocities in the direction of impact between two touching balls are uncorrelated.

### Analysis

- Fix $\left\{a_1, \dots, a_n \right\} \subset \mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2\pi]$. The probability $\mathbb{P}(a_1,\ldots,a_n)$ that $\cos(a_1 x), \dots, \cos(a_n x)$ are either all positive or all negative is non-zero since $\cos(a_i x) \sim 1$ for $x$ in a neighborhood of $0$. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that $\mathbb{P}(a_1,a_2) \geq 1/3$ with equality if and only if $\left\{a_1, a_2 \right\} = \gcd(a_1, a_2)\cdot \left\{1, 3\right\}$. We prove $\mathbb{P}(a_1,a_2,a_3)\geq 1/9$ with equality if and only if $\left\{a_1, a_2, a_3 \right\} = \gcd(a_1, a_2, a_3)\cdot \left\{1, 3, 9\right\}$. In this presentation, we sketch the main ideas behind this result and briefly discuss the surprising behavior that shows up when we consider more than three cosines.

### Algebra

- We define the prime graph $\Gamma (G)$ of a group $G$ as follows: The vertices are the prime divisors of $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there exists an element of order $pq$ in $G$. A complete classification of prime graphs of finite groups remains elusive, yet progress has been made for some families of groups. For example, it has been shown that a graph is the prime graph of a solvable group if and only if its complement is triangle-free and 3-colorable. We hope to find similar classifications for groups with nonabelian composition factors. To that end, we make two definitions. First, given a nonabelian simple group $T$, we say that a group is $T$-solvable if its composition factors are either cyclic or isomorphic to $T$. Second, we say that a simple group is $K_n$ if its order has exactly $n$ prime divisors. The prime graphs of groups with $K_1$ or $K_2$ composition factors are well-understood because such groups are always solvable. Last year, a team of undergraduate researchers classified the prime graphs of $T$-solvable groups for all $K_3$ groups $T$. Now, we present further classification results for $T$-solvable groups, this time selecting $T$ from a large family of $K_4$ groups.