Welcome to the Math and Stats Independent Comps Presentations for Fall 2020! This term students are recording comps presentations that are viewable by a prerecorded video. Please refrain from posting these videos to any other online resource. Independent comps students will be hosting a Q&A session by Zoom on November 10 and 12 at the times listed below. Q&A questions submitted by 2 pm the day of the presentation will receive full consideration. The Zoom link will be emailed prior to the event. We hope you enjoy the comps presentations.

## Tuesday, November 10, 2020

Time: 4:00-4:20 pm
Title: Spatial Data Analysis with Applications in Epidemiology
Speaker: Molly Potter

Comps Presentation Video
Abstract: How do epidemiologists identify spatial patterning in the occurrence of diseases in an effort to understand how illnesses spread? Geographic trends in the spread of diseases can be understood with data that identifies the presence of an illness of interest in specific locations, along with information on explanatory variables that could influence the number of infected individuals. Spatial correlation, specifically autocorrelation, can be measured through the comparison of indexes to permutation distributions as well as through conditional autoregressive modeling. Spatial autocorrelation can be defined as the systematic correlation of multiple observations in a variable of interest occurring in close proximity. Measuring this type correlation requires us to rethink the way we typically view data, where a case’s observed measurement in a variable of interest must be thought of in unison with the observed values of the cases that are nearby. My talk will present a case study involving the incidence of lung cancer in the counties of Pennsylvania to investigate how to work with areal spatial data, and in turn, how spatial autocorrelation is measured.

Time: 4:20-4:40 pm
Title: Generalized Additive Models
Speaker: Jaylin Lowe
Comps Presentation Video
Abstract: How do we model data that is non-linear? In linear regression, we assume that the underlying relationship between our response variable and our explanatory variables is linear. Along with related techniques, generalized additive models provide a framework for modeling data when this linearity assumption is violated. In this talk, I will discuss how generalized additive models extend the generalized linear model framework to fit data that is not linear, while still maintaining the additive nature of linear models. Generalized additive models add the extra layer of complexity necessary to model non-linear data by considering the response variable to be the sum of complex functions of explanatory variables, rather than just the weighted sum of the explanatory variables alone, as in linear regression. There are a variety of different techniques used to create these functions of explanatory variables, including polynomial regression, local regression, regression splines, and smoothing splines. I will go into detail about these methods and how they can be combined to form generalized additive models, in addition to illustrating how generalized additive models can be used to analyze data about US colleges and universities.

Time: 4:40-5:00 pm
Title: Lie Groups and Particle Physics: Representing Physical Symmetry with Group Theory
Comps Presentation Video
Abstract: Symmetry lies at the heart of today’s theoretical study of particle physics. In this talk, we present foundational mathematics for understanding physical symmetries. We start from basic group theory and explain how differential symmetries can be represented through Lie groups and their generators’ algebra. We then introduce several important Lie Groups such as the SU(2) Group and Lorentz/Poincare Group and talk about their physical meanings. Finally, we compare particle theory with field theory and apply Noether’s Theorem to a quantum relativistic field. We believe that the materials cover here will prepare undergraduates for future studies in mathematical physics.

Time: 5:00-5:20 pm
Title: The Gaussian Moat Problem
Speaker: Allie Clark
Comps Presentation Video
Abstract: In one dimension – on the real number line – it is impossible to “take a walk” from zero to infinity only stepping on prime integers, at least if your “steps” have to be of bounded size. The Gaussian Moat Problem asks whether such a thing is possible in the two dimensions, using the complex plane. Specifically, it asks whether or not it is possible to walk from the origin to infinity taking steps of bounded size on the Gaussian primes. Though the problem is still unsolved, I will be describing its history in more detail, along with demonstrating the proof of one “moat” and explaining the implications.

Time: 5:20-5:40 pm
Title: p-adic Numbers: What happen when we have another absolute value
Speaker: Eric Shao
Comps Presentation Video
Abstract: We define the field of real number from rational number with Cauchy sequences. The distance between two numbers are initiatively the absolute value of the difference. What will happen if we define another absolute value? In 1916, Russian mathematician Alexander Ostrowski tell us there can only be two kinds of absolute value besides the trivial one on rational number. We will figure out the other kinds of absolute value with prime number and proof his theorem. With this more “general” absolute value, we will construct the field of p-adic number just as we construct real number out of rational number. Then we can explore the topology of this field.

## Thursday, November 12, 2020

Time: 4:00-4:20 pm
Title: Differential forms, their discretization, and applications to computer graphics
Speaker: Ross Grogan-Kaylor
Comps Presentation Video
Abstract: Differential geometry is in many ways a thorough investigation and reorganization of the integral theorems of “vector calculus”. Remember div, grad, and curl? Green’s theorem? The divergence theorem? Maybe Stokes’ theorem? These ideas are beautifully generalized by the notion of differential forms, which are algebraic objects invented to be integrated over “multidimensional surfaces”. We will briefly introduce differential forms and present the generalized Stokes’ theorem, which is a very general version of the fundamental theorem of calculus. The generalized Stokes’ theorem is deceptively simple in appearance; it looks simpler than the theorems it generalizes!

After introducing differential forms, we ponder a central theme of calculus: using limiting processes to produce smooth objects from finite ones. Can we do anything interesting if we reverse this process? That is, what if we start with a smooth theory and then discretize? We will present how differential forms, when discretized, can be utilized to solve problems in computer graphics. Namely, we will show how discrete differential forms can be used to “smooth out details” on a surface and to wrap a two-dimensional map around a globe in an angle-preserving manner.

Time: 4:20-4:40 pm
Title: The Banach-Tarski Paradox
Speaker: Elwood Olson
Comps Presentation Video
Abstract: The Banach-Tarski Paradox, a startling consequence of the Axiom of Choice, says that a sphere can be cut up into a finite number of pieces and rearranged into two new spheres identical to the original using only distance-preserving motions. In my presentation, I will exhibit a proof of this deeply counter-intuitive result, exploring tools from algebra, set theory, and geometry along the way. First, I will formalize the algebraic structure underpinning the paradox with a group theoretic object called paradoxical actions. Next, I am going to use the Axiom of Choice to demonstrate how this algebraic structure can be grafted onto the geometry of a sphere’s rotations. Finally, I will demonstrate a method called proof-by-absorption, which I will use to prove the Banach-Tarski Paradox itself.

Time: 4:40-5:00 pm
Title: Exploring Eventual Extinction in Branching Processes
Speaker: Jack Moran
Comps Presentation Video
Abstract: In 1873, Sir Francois Galton posed a question regarding the survival of Victorian aristocratic surnames over many generations. “Assume that surnames are inherited from the father. If all adult males in a population independently have zero male children with probability a0, one male child with probability a1, and so on with k male children according to probability ak, then what is the probability that a given surname will become extinct after n generations? If the surname is not extinct, how many individuals will hold it?” The study of branching processes emerged from this line of questioning. Generally, branching processes are useful in studying the survival of a population and are commonly used to model the spread of disease, ecological populations, and cell mutations. In this talk, we will build up intuition about the long-term behavior of a branching process given the distribution of its offspring and derive a probability of eventual extinction for a process using probability generating functions.

Time: 5:00-5:20 pm
Title: Lighting the Lamp: Calculating Individual Player Contribution in Hockey
Speaker: Dan Clipper
Comps Presentation Video
Abstract: Plus-minus is a traditional metric used in hockey to evaluate a player’s overall contribution. It gives the difference between the number of goals scored and the number of goals conceded by a player’s team while that player was on the ice. What does this tells us? Well, pretty much nothing, other than a player happened to be on the ice when a goal was scored. This is what’s called a marginal effect; it doesn’t take into account the influence of the other players on the ice. Is there a better way to evaluate players? Adjusting a metric like plus-minus to account for the other players would instead give us a partial effect: a player’s contribution regardless of who else is on the ice. This talk will discuss some of the issues with traditional metrics, detail a model to calculate better ones, and reflect on what this all means for hockey and other sports.

Time: 5:20-5:40 pm
Title: Are personality traits universal? Exploring the Cross-culture Validity of Big Five Personality Traits Using Factor Analysis
Speaker: Changlan Wang
Comps Presentation Video
Abstract: Factor analysis is a frequently used statistical method to develop survey in psychology, marketing and social sciences in general. The main purpose of factor analysis is to describe variability among observed variables in terms of a potentially lower number of variables, which are called factors. Factor analysis was first used by the English psychologist Charles Spearman as a method to investigate bifactor theory of human intelligence (also known as general intelligence g). In this project, I used factor analysis to explore the cross-culture validity of the widely-known Five Factor Model of personality traits in a non-western context.