A permutation is a list where order matters. Despite this humble definition, permutations offer a wealth of beautiful mathematics that can be applied across scientific disciplines. Starting simply, with lists of numbers, we'll discuss how smaller permutations can be embedded into larger permutations along with a host of interesting counting problems with connections not just to mathematics, but to computer science, chemistry, and more.

A positive integer that is divisible by the sum of its digits is called a harshad number (or Niven number). This simple concept leads to many interesting questions and some surprising answers. In this talk, I will invite you to think about these numbers and some variations; I will share with you a variety of results and show you how to prove some of them; and I will leave you with a number of open (unanswered) questions, some of which you may decide to explore.

Lean is a proof assistant created by Leonardo De Moura in 2013 as a Microsoft Research project. Its mathematical library, mathlib, is a collective endeavor to establish a comprehensive repository of formalized mathematics within the Lean proof assistant. In an effort to further expand the extensive mathlib, we worked on defining the orthogonal and special orthogonal groups in Lean. Then, with these definitions as a foundation, we were able to prove lemmas and theorems that were previously absent from the mathlib.

The theory of factorization is a fundamental topic in number theory, and understanding how factorization within an object behaves gives a rewarding perspective into its underlying structure. In this talk, we begin with a self-contained introduction to non-unique factorization, through the study of several classes of semigroups, with a focus on investigating the factorization length of iterated powers of elements. Building on results of previous summer research at San Diego State University, we introduce a generalization of factorization length. Using this new computation, we present results ranging from asymptotic behavior of iterated powers to complete, quasi-polynomial characterizations for the factorization length of elements.

We say that a curve \(C\) in \(P^2\) defined over \(F_q\) is transverse-free if every line defined over \(F_q\) intersects \(C\) at some point with multiplicity at least \(2\). Bounds on the asymptotic proportion of transverse-free curves were previously derived by Asgarli and Freidin in the smooth case. We tackle the general case and show that there is a special class of curves, which we name trivial blocking curves, that are always transverse-free. In particular, we find a lower bound on the size of this class of curves, and show that this bound is asymptotic to the true proportion of transverse-free curves as \(q\) approaches infinity, that is, nearly all transverse-free curves are trivial blocking curves for sufficiently large \(q\).

In this talk we study some algebraic aspects of collections of linear operators between finite dimen- sional complex vector spaces of modular forms. In particular, we’ll address a series of conjectures posed by Huber, Alaniz, and others predicting the form of the determinants of classes of integer-valued matrices associated with the aforementioned linear operators.

With recent advancements in the field of nanotechnology, there has been increasing interest in self-assembling nanostructures. These are constructed through the process of branched junction DNA molecules bonding with each other without external guidance. Using a flexible tile-based model, we represent molecules as vertices of a graph and cohesive ends of DNA strands as complementary half-edges allowing the molecules to bond with each other. Due to the unpredictability of DNA self-assembly in a laboratory setting and the risk of undesirable products being incidentally constructed, predicting what structures can be produced from a given list of components, referred to as a “pot of tiles,” is useful but has been proven NP-hard. This project introduces an algorithm to computationally generate and visualize at least one valid graph and for smaller cases, all non-isomorphic graphs, given a pot of tiles. By adjusting a number of construction parameters, we can produce graphs of various orders and proportions of tiles.

Chronic thromboembolic pulmonary hypertension (CTEPH) is a fatal but curable disease afflicting the pulmonary arteries. Diagnoses include a mean pulmonary arterial pressure (mPAP) greater than 20 mmHg and unresolved lesions in the pulmonary arteries. CTEPH is widely underdiagnosed since its symptoms (dyspnea, exercise intolerance, malaise) are common in many other diseases, and it is detected only through CT imaging and right heart characterization. One treatment for these patients is balloon pulmonary angioplasty (BPA), where stents are inserted into blood vessels and inflated to increase blood flow past embolisms. Currently, there is no systematic way to determine which lesions to treat to maximize outcomes: increasing perfusion and decreasing pressure. To remedy this, we devise a personalized one-dimensional (1D) fluid dynamics model predicting blood pressure and flow in the pulmonary arteries. Using this model, we will compare outcomes in control and patient networks. To simulate treatment, we will remove different combinations of lesions in personalized networks generated from CT images to optimize prognosis.

Using 3D Slicer, we generate 3D volumetric surfaces (segmentations), and from these, using the Vascular Modeling Toolkit (VMTK), we extract centerlines along each vessel and connect them in a labeled tree. Nodes along the centerline each have an associated radius and coordinates in 3D space, and the connection of nodes along a vessel gives us vessel length. Junctions form when nodes intersect, and we represent a network of vessels through a connectivity matrix. To find a representative radius, we average the radii of relevant nodes, identified using our change point algorithm. This algorithm determines vessel radii and its uncertainty and corrects junctions between vessels to be more physiologically accurate. We pruned networks until they had the same number of arteries to standardize networks. Then, we attach asymmetric binary trees at the end of each terminal vessel to extend our network past resolution limits in CT scans. We solve a 1D fluid dynamics model in this network, predicting blood pressure and flow in each vessel.

Preliminary results predicting mean pulmonary arterial pressure and flow in five segmentations from a healthy control’s pulmonary arteries show that the size of the tree matters, and, using ANOVA analysis, we found that vessel radii and length differ significantly between segmentations; however, sampling from the radius estimates for each vessel provides reliable predictions of pressure and flow. Segmenting CTEPH patients reveals multiple lesion types per patient, including ring lesions, total occlusions, and tortuous vessels. Future work includes generating CTEPH networks and conducting fluid dynamics simulations in CTEPH geometries. These in-silico treatments and simulations mitigate the need for invasive and expensive scanning and provide insight into optimal treatment plans for physicians.

Emily Payne and Abigail Martinez extend Euler's "odd=distinct" partition identity to a two-variable identity which considers the number of times parts divisible by natural number \(m\) may appear.

While an analytic proof is evident, we employ the part-frequency tables of William Keith to combinatorically prove and further generalize this extension.

Our project aims to use machine learning exploratory data analysis to find unseen patterns in university student data that would predict student performance and help advisors to promote student success. We will discuss our data and the testing of several different models - the random forest model being most accurate. The main issues in our data are being widespread missing information and a variety of staggered information.

We will briefly discuss hyperbolic space, one of the most important kinds of non-Eulidian geometry. We will demonstrate how to visualize and interact with hyperbolic space by implementing 4 x 4 transformations to represent isometries of hyperbolic space in WebGL, a web-based graphics library.

We explore the relationship between average daily income per capita and happiness scores for various countries around the world. We produce and analyze linear and nonlinear models of the data over the years 2008 - 2022 that can best explain the existence of a relationship. Accordingly, we compare a subset of countries with distinctive models in preparation to research events within those countries that could have influenced their happiness scores or average income.

The Golden Ratio of mathematics has been seen all throughout nature and within the human features. In art, the Golden Ratio is especially used in portrait making as it is said the face is to follow the ratio. We will present the golden pentagram and its uses in portraiture. To do so, we will create our own pentagrams of specific subjects and compare the results with the Golden Pentagram. We are working towards a mathematical model of this process that can be used on any subject.

We consider historical initial public offering (IPO) data from 2000-2020. The model will analyze over 3000 stock tickers. We will determine general trends and examine possible investment strategies.

Wolves have a long and complex history in Yellowstone National Park. In 1995. They were the apex (highest) predator. Because of this human intervention however, the ecosystem as well as the relations of multiple species was thrown very out of balance. In this project we analyze 3 different species in Yellowstone National Park: wolves (the apex predator), elk (prey/consumer), and berry-producing shrubs (resource). Further, we will divide up the predator based on their maturation by dividing them into two different groups: wolf pups and adult wolves. We implement a published model combining Omnivory and competition models, to create one system of differential equations.

Fractional (diffusion) partial differential equations (FPDE) can model anomalous diffusion using fractional order stochastic processes instead of Brownian motion used in the regular diffusion equation. In this work, we present a new numerical integration method to approximate solutions to (FPDE). The implicit trapezoidal, a time-stepping method, allows us to solve numerically the PDE through the use of composite Simpsons, an integrating technique. Composite Simpson allows us to approximate FPDE numerically using a Wright-type transformation. In return, this yielded solutions to distinct homogenous and non-homogenous fractional heat and wave equations with their corresponding initial and boundary conditions.

Computer algorithms are utilized more and more each year in order to solve complicated math problems. We show how one can obtain good approximations to solutions of nonlinear ordinary differential equations using some popular numerical methods, including Euler, Heun, and Runge-Kutta.

Two graphs \(G\) and \(H\) are distance cospectral if their distance matrices have the same spectrum. We use block similarity matrices to show that G and H remain distance cospectral after coalescing (gluing) graphs onto certain subsets of the vertices if the similarity matrix commutes with the all ones matrix.

Identification of sleep disorders is essential as they can have dangerous consequences on human health. This can prevent people from having a quality lifestyle. It is known that certain sleep disorders are difficult to treat, though some are easy to manage. This study reports the findings of predicting people's sleep disorders using a publicly available dataset. We employ two Logistic Regression models to determine whether subjects have developed a sleeping disorder. Based on the AIC value and ANOVA test, out of the full and the reduced models, the latter performs better than the former. According to the experimental findings, age (p=0.013), quality of sleep (p<0.01), and diastolic blood pressure (p<0.01) significantly impact sleeping disorders.

In December 2021, President Biden’s target through the Bipartisan Infrastructure Law was to have 50% of electric vehicle sale shares by 2030 and zero-net emissions by 2050. 21 months have now passed since this target was made. In a period of ten years, for this target to be reached, we should see the percentage of electric vehicles to gasoline-powered vehicles increase by approximately 10% each year. According to data collected from the U.S. Department of Energy, currently from 2021 to 2022, the number of electric to gasoline-powered vehicles has increased by only 0.41%. The purpose of this research is to look at the reasons for the slow transition from gas-powered to electric vehicles. Using data gathered from the United States government reports and conducting a survey based on a representative sample including differences in age, gender, and economic standing, we will analyze the factors that hinder the transition from gas-powered to electric vehicles and discuss recommendations to speed up this transition.

The candy sharing problem is as follows: there are n children sitting in a circle. They are playing a game where they each start with an even number of candies. Every round has the players pass half of their candies to the person on their left. If they have an odd number of candies by the end of the round, they will be given one candy, so that they have an even number of candies. The game ends when everyone has the same number of candies. Our goal is to understand the mathematics of this game. In doing so, we look at a game with 3 players and a game with 4 players. We fully characterize the possible combination of candies in a single round. We also find the third and second to last states of candies in a game with 3 players and the penultimate sequence of candies in the game with 4 players.

In the last few decades, concerns regarding air pollution have led to many new laws and regulations being put into place to mitigate the effects of pollution on the environment and humanity as a whole. This paper analyzes several decades' worth of ground-level ozone readings in six major cities/regions in Texas, using data from the United States Environmental Protection Agency (EPA). These regions include the Austin-Round Rock area, Corpus Christi, the Dallas-Fort Worth-Arlington area, El Paso, the Houston-The Woodlands-Sugar Land area, and the San Antonio-New Braunfels area. We identify trends in these readings using the Jonckheere-Terpstra test.

Elliptic curves are crucial objects in mathematics which bridge many different areas, namely analysis, algebraic number theory, and cryptography. This talk will be a gentle introduction to Elliptic Curves highlighting some interesting results. The main goal will be to develop an intuition for finding rational solutions of polynomials, see a geometric demonstration of elliptic curve addition, and understand the importance of Elliptic Curves.

In the vast and ever-expanding realm of modern science and engineering, applied mathematics remains ever-present, weaving seemingly disparate fields into a consistent combination of innovation. This presentation delves into the impact and intersections of applied mathematics across diverse disciplines in my research. Drawing from rigorous research in robotics, aerospace engineering, ECG signal processing/machine learning classification, and nanomanufacturing, we will embark on a journey that showcases the versatility and indispensability of mathematical frameworks. Each case study illuminates the intricate spin of equations, algorithms, and principles that drive breakthroughs and advancements. By demystifying these intersections, this discourse aims to highlight the boundless potential of applied mathematics, asserting its role as the mastermind of modern technology. Join in as we navigate the phenomenon that shapes our understanding, pushing the boundaries of what's conceivable and achievable in the interconnected world of science and technology.

A perfect number, a well-known concept which has been studied since ancient times, is a natural number that is equal to the sum of its positive proper divisors. As a generalization of perfect numbers, a natural number n is said to be a Zumkeller number, or an integer-perfect number, if the set of its positive divisors can be partitioned into two subsets of equal sums. The concept of Zumkeller numbers was first introduced by LeVan in 1987 and then popularized by Zumkeller, who published this class of numbers on the On-Line Encyclopedia of Integer Sequences (OEIS) in 2003. Since then, several authors have studied Zumkeller numbers. In our research, we continue this study and provide several new results regarding Zumkeller numbers in arithmetic progressions and sums involving Zumkeller numbers.

This project aims to provide the drone pilot with another degree of control, by mounting the Attitude and Heading Reference System (AHRS) to the FPV goggles and using the attitude output as an input to the gimbal control via telemetry radio channels providing the pilot with robust intuitive control of the camera. AHRS is a device which uses an assortment of individual sensors as inputs into a Kalman Filter. Specifically, the AHRS outputs a 3 Space vector, altitude, and compass heading. The Kalman Filter is an optimal estimation algorithm, which uses this continuous data to output the current state based on current measurements and prior knowledge of state, in real time. In many cases drone cameras are static or restricted to manual tilt adjustment. However, this current work aims to change that and will have applications for many areas including drone inspection, search and recovery, and defense.

Predator-prey systems have been utilized to explore and model population dynamics. The Lotka-Volterra ODE models the periodic relationship between predator and prey populations. We modified the model to include both carrying capacity and minimum prey size. In this research study, the modified versions of the Lotka-Volterra model were simulated in R. We explored parameters to demonstrate Allee effects and climate change over time on the predator-prey systems. In our preliminary analysis, the magnitude of the climate change parameter played a key role in the relationship of the predator-prey systems. At low levels of climate change, both predator and prey populations displayed an oscillatory behavior. When climate change levels were increased, the population of the prey survived in an oscillatory manner with no risk of extinction, yet the predators’ population went extinct. Our analysis suggests that the interplay of climate change and Allee effects impacts the behavior of predator-prey populations.

We analyze the coordinates, right ascensions, and declinations of various star clusters in the Milky Way galaxy using the William E. Harris 2010 data set. We model the center of the Milky Way and calculate the average distance to the center from our solar system in lightyears.

We seek to characterize the \(3\)-adic valuations of the family of functions \(x^2 + a\), and what values of \(a\) in one of three forms determines the valuation to be. We partition the family of functions into three conjectures, depending on the three forms of \(a\). Finally, we prove one of the three conjectures, that the valuation of \(x^2 + 3a + 1\) is \(0\) for any natural number \(x\).

Iterative methods are useful when successive approximations can generate solutions when they converge to exact solutions. Presenters will be discussing a viable alternative algorithm that can provide solutions with sufficient accuracy with fast convergence.

Presenters will show examples of tests for convergence and explore the statistics of convergence under different conditions. We will also be discussing many problems that can occur and create complexity for iterative methods.

We studied the Kuramoto model to create modifications to transform the natural frequency as a function with respect to time. We constructed four models using different weights of the prior velocities. In comparison of the models, we found the Exponentially weighted intrinsic function exerted efficiently in faster and more steady synchronization.

Employing tools from graph theory and linear algebra, we model the biological process of the creation of nanostructures from self-assembling DNA complexes. We represent k-armed branch junction molecules with tiles which are vertices in a graph with half-edges. The half-edges depict the cohesive-end types of a DNA strand. We aim to determine the minimum number of tiles and cohesive-end types necessary to form the complete complex of a given multi-dimensional graph structure. The problem of modeling DNA self-assembly is particularly challenging when considering graph families which change in multiple dimensions. In this research, we present the minimum number of tiles and cohesive-end types necessary to create the stacked book graphs, the square lattice graphs, and the Mongolian tent graphs, under different laboratory constraints.

Just as any compact, oriented, connected \(3\)-manifold can be uniquely decomposed into two \(3\)-dimensional handlebodies via a Heegaard splitting up to Heegaard moves, any compact, oriented, connected \(4\)-manifold can be decomposed uniquely into three \(4\)-dimensional handlebodies up to a natural stabilization operation. In particular, we can ask that the pairwise intersection of these 4D handles are Heegaard splittings along the closed surface sitting in the triple intersection; if this surface is of genus \(g\), we call this a \(g\)-trisection of the given \(4\)-manifold. If we allow \(n\geq 3\) 4D handlebodies with the same intersection conditions, we call this a \((g,n)\)-multisection. These decompositions allow us to translate questions on \(4\)-manifolds into questions about curves on surfaces, namely the curves specifying the gluings which define a multisection. This process simplifies the computation of classical algebraic invariants like homology and intersection forms and gives rise to a number of natural invariants like minimal genus. We concentrate specifically on \((2,n)\)-multisections whose pairwise intersections are Heegaard splittings for integer homology spheres and what restrictions this family of \(4\)-manifolds has on its intersection forms. This talk will focus mainly on the background to this exciting new method to studying \(4\)-manifolds.

Let \(p(n,3)\) be the number of partitions of \(n\) into at most three parts. We define \(p(n’,3)\) as a partner of \(p(n, 3)\). A crank is a partition statistic that witnesses a partition congruence by distributing partitions into equally sized subclasses. We investigate the congruence of linear combinations of \(p(n, 3)\) and \(p(n’, 3)\equiv 0\mod 3\). We show that the partition statistic “sigma,” computed by the number of parts of size \(1\) minus the number of parts of size \(3\), is a crank witnessing this congruence under specific conditions. We conjecture that modifications for \(n\) and its partner \(n’\) can allow the partition statistic to be a crank witnessing many more congruences.

Stochastic ordering of probability distributions holds various practical applications. However, in real-world scenarios, the empirical survival functions extracted from actual data often fail to meet the requirements of stochastic ordering. Consequently, we must devise methods to estimate these distribution curves in order to satisfy the constraint. In practical applications, such as the investigation of the time of death or the progression of diseases like cancer, we frequently observe that patients with one condition are expected to exhibit a higher likelihood of survival at all time points compared to those with a different condition. Nevertheless, when we attempt to fit a survival curve based on real-world data, this anticipated behavior may not always be seen. Therefore, it becomes crucial to estimate these curves to provide a more accurate representation of the true survival times for patients under different conditions. To address this challenge, we harness the insights of various statisticians and adapt their methodologies for one-sample and two-sample cases to the more complex scenario of a three-sample case. In this case, we work with data obtained from three distinct populations for several kinds of cancer. Given the inherent complexity of such data, it is highly likely that the empirical survival functions derived from it will not conform to stochastic ordering constraints, necessitating the estimation process. This study actively investigates four different estimators applied to data representing the relative survival rates of various racial groups affected by eight different types of cancer. Ultimately, our goal is to actively determine which, if any, of these estimators perform the best in terms of Bias and Mean Square Error. This active analysis will aid in actively selecting the most suitable estimator for future investigations involving survival data.

In 2001, Zagier studied various properties of a \(q\)-hypergeometric series \(F(q)\), originally introduced by Kontsevich in 1997 which only makes sense for \(q\) a root of unity. In particular, Zagier showed that \(F(q)\) satisfies a “strange identity” which leads to its quantum modular transformation properties. In fact, Andrews and Sellers (2016) considered the coefficients in the \((1 − q)\)-expansion of \(F(q)\), also known as the Fishburn numbers, and obtained several congruences. Motivated by this, there has been substantial interest in generalizing these congruences for similar expansions of \(q\)-hypergeometric series in the Habiro ring that satisfy a strange identity. In this work, we expand upon the previous work to two non-Habiro-ring elements considered by Andrews, Jiménez-Urroz and Ono (2001) that also satisfy strange identities. We show that the coefficients (appropriately normalized) in the \((1 − q)\)-expansions of these series give rise to similar congruences.

We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by a decreasing path (attainability problem), and we show that this can be done by a finite application of "pull-in" moves (bang-bang problem). In the case of triangles, this problems was investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.

Supraventricular Tachycardia (SVT) encompasses irregularly fast heart rhythms originating from the atria. Although SVT seldom directly leads to death, its various forms, including atrial fibrillation, stand as the primary cause of strokes and heart attacks worldwide. SVT also contributes significantly to preventable heart failure cases through tachycardiomyopathy and can trigger myocardial infarction by increasing oxygen demand while reducing cardiac output.

While medication plays a crucial role in SVT management, cardiac ablation has emerged as the most effective long-term solution. In this procedure, an electrophysiologist employs tiny catheters to create a 3D map of the heart's electrical activity. Advanced computer software helps identify the source of the irregular heartbeat, allowing targeted radiofrequency ablation of specific areas to restore the heart's normal rhythm.

Despite significant advancements in cardiac ablation over the past two decades, the causes and optimal ablation methods for the most prevalent and problematic cardiac arrhythmia, Atrial Fibrillation (Afib), remain a modern-day mystery. Understanding the underlying mechanisms for Afib remains limited. To address this, we conducted an in-depth review of current literature and utilized a computational model of the Left Atrium to explore theories related to Afib's cause and mechanisms. Through these efforts, we aim to shed light on this challenging arrhythmia and enhance treatment approaches.