Welcome to the Math and Stats Group Comps Presentations for Spring 2020. This term students have recorded comps presentations that are viewable by a prerecorded video. Please refrain from posting these videos to any other online resource. We hope you all enjoy the comps presentations.

## Tuesday, May 19 from 4:00-5:00 pm

**Time: 4:00-4:20 pm****Bootstrap Confidence Intervals for OLS Regression**

Comps Advisor: Professor Laura Chihara

Students: Andrew Lin, Nobuaki Masaki, Colin Pi, Matt Thill, Yihuang Wu

Comps Presentation Video

Abstract: Classical t confidence intervals for the slope and intercept in OLS regression may not capture true parameters at the intended frequency in the presence of heteroscedasticity or outliers. For our project, we first looked at techniques such as resampling cases and residuals to obtain bootstrap distributions for estimated regression slopes and intercepts. Then, we explored the BCa confidence interval in addition to the bootstrap percentile confidence interval used to make inferences about the slope and intercept from bootstrap distributions. Using simulation, we compared across samples drawn from various populations the coverage of 95% confidence intervals for the slope and intercept obtained from the techniques we explored to test their performance and see whether they were robust to model violations such as heteroscedasticity.**Time: 4:20-4:40 pm****A Dynamic Exploration of the Mandelbrot Set**

Comps Advisor: Professor Rafe Jones

Students: Ozzy Houck, Dylan Kempton, Milena Silva, David White

Comps Presentation Video

Abstract: The Mandelbrot set is one of the most famous (and beautiful) objects in mathematics. This immensely complicated set has its roots in a simple question: what happens when you repeatedly apply a complex quadratic polynomial to a complex number? We use tools from complex dynamics and analysis, including the Schwarz Lemma, the Attracting Periodic Orbits Lemma, and the Fatou-Julia Lemma, in order to describe the Mandelbrot set. Along the way, we cover fun topics such as periodic cycles, basins of attraction, and filled Julia sets. We show that even small changes in our choice of starting point can have dramatic effects on the dynamical outcome of the system, and that even simple problems can give rise to beautiful complexity. **Time: 4:40-5:00 pm****Wiring the Universe**

Comps Advisor: Professor Owen Biesel

Students: Emma Qin, Will Schwarzer, Peter Sparks, Elizabeth Zhu

Comps Presentation Video

Abstract: In this interdisciplinary project, we show how wiring diagrams, a concept from applied category theory, can be used to produce models in a diverse range of subjects. We discuss examples ranging from truth tables in electrical circuits to propositional logic and from problems in geology to economic models.

## Thursday, May 21 from 4:00-5:00 pm

**Time: 4:00-4:20 pm****A Hazardous Analysis: Using Survival Methods to Understand Factors Affecting the Age of First Drug Use**

Comps Advisor: Professor Tom Madsen

Students: Natalie Maurice, Tanvi Mehta, Matteo Pellizzer, Lewis White

Comps Presentation Video

Abstract: Understanding and modeling the time it takes for a specific event to take place is a valuable statistical tool. In situations where complete information of the time variable is missing due to the observation period ending before the event takes place or other such instances, survival analysis provides methods to analyze this data, making use of all of the available information. One interesting application for this theory is the time until individuals first try certain drugs. In this project, Cox proportional hazard models and Weibull accelerated failure time models were used to analyze how different demographic factors such as gender, race, education level, economic status, perceived risk of the substance, and age of first use of other substances affect the age at first use of marijuana, cocaine, and heroin. Largely, these two models had similar results for each of the three substances, and diagnostics suggest that the assumptions of both are reasonable. Impacts of these six factors on the hazard of trying a substance varied between the three substances. Further work could look into the varying effects of age over time, other demographic characteristics, and additional substances to see how consistent these results remain.

**Time: 4:20-4:40 pm****Am I Ready to Learn “Real” Math Now? A Presentation About Northfield’s First Math Circle**

Comps Advisor: Professor Deanna Haunsperger

Students: Jackie Chan, Tenzin Kunsang, Elisa Loy, Fares Soufan, Taylor Yeracaris

Comps Presentation Video

Abstract: Do school students need to wait until college-level mathematics or statistics before they can experience the true joy of mathematical discovery? By starting a Math Circle in the Northfield Middle School, we answered this question with a resounding “NO.” Our Math Circle, which we ran weekly during the Fall and Winter terms, gave students a chance to approach math as mathematicians do: creatively and freely making mistakes, making discoveries, and asking new questions, all on their own terms. By letting students take the lead, we let them be the owners of their own mathematics, and helped them to experience the joy of mathematical exploration. Along the way, we exposed them to several of our favorite math topics, most of which are outside the usual public school curriculum; these included combinatorics, graph theory, probability, and cryptography. In our comps talk, we will discuss the mathematics, motivations, methods, and—yes—madness of our Middle School Math Circles. Come on by!

**Time: 4:40-5:00 pm****Homological Algebra**

Comps Advisor: Professor Mark Krusemeyer

Students: Oscar Smith, Siang Wongrattanapiboon

Comps Presentation Video

Abstract: Homological algebra is the study of how to associate sequences of algebraic objects such as abelian groups and modules to topological objects in a general algebraic setting. Homological algebra began to be studied as a branch of topology in the 1800s, and became an independent subject at the end of the 19th century, chiefly by Poincaré and Hilbert. One motivation for this subject is the study of chain complexes, a useful concept in several areas of mathematics including abstract algebra and differential geometry. In this talk, we will introduce R-modules, exact sequences, categories, and functors, and show how they can allow us to extract information contained in chain complexes. Along the way, we will also demonstrate a (super cool!) method of mathematical proof called diagram chasing.