Spring 2023 Program

Meet the Mentors

Nick Backes

PhD student in Math at Rutgers University

Description: The p-adic numbers are a wonderful number system where every point of a circle is the center and every triangle is isosceles. When we add p-adic numbers, we carry the 1 to the right, instead of to the left. And in the p-adics, the integers are all small. In fact, the largest p-adic integers have size 1.

The real numbers extend the rational numbers by "filling in the gaps" between rational numbers. For example, consider this convergent sequence: 3,  3.1,  3.14,  3.141, 3.1415, etc. Each term is a rational number, but the limit, pi, is irrational. We say the real numbers are a completion of the rational numbers.

But what if we fill in the gaps in a different way? What if we change the meaning of convergence, to allow different sequences to converge? This is where p-adic numbers come from. They are a different completion of the rational numbers. It turns out that the real numbers and the p-adic numbers are the only completions of the rational numbers.

In this project, we will start from scratch, construct the p-adic numbers, and explore some of their amazing properties. 

Suggested Book: Chapter 1 of p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz.

Prerequisites: Familiarity with proof-based math is necessary. Familiarity with any of the following will help, but none are necessary: modular arithmetic, metric-space topology on the real numbers, Cauchy sequences, and the basics of topological spaces (open sets, closed sets, compact sets, and continuous functions).

Krishan Canzius

PhD Student in Math at the University of Pennsylvania

Description: Sets are ubiquitous throughout mathematics: we study sets of numbers, sets of vectors, sets of functions, and “sets with structure." However, we rarely study sets in their own right, instead relying on our intuitive understanding of how sets should work. This strategy works well when dealing with finite sets, but our intuition can betray us when working with infinite sets.

What does it mean for one infinite set to be larger than another infinite set? Are there more natural numbers or rational numbers? How do these compare with real numbers? Is there a largest set? If you combine two infinite sets, will the whole be larger than both of the parts? What if you combine countably many infinite sets instead?

In this project, we will answer these questions and many more. We will study cardinal numbers, cardinal arithmetic, ordinal numbers, and ordinal arithmetic. If there is time, we can also begin a study of axiomatic set theory in order to rigorously define what a set is.

Suggested book: Naive Set Theory by Paul Halmos

Prerequisites: Some familiarity with proof based math.

Marlin Figgins

PhD Student in Applied Math at the University of Washington

Description: Mathematical models of infectious diseases have been essential in developing our modern understanding of how and why infectious diseases continue to spread. There are various ways of modeling infectious diseases using ordinary differential equations, discrete dynamical systems, stochastic processes, agent-based models, and machine learning. In this project, we will study models of infectious disease and learn how to use mathematics to describe and simulate infectious disease systems. We will then use these ideas to learn how to fit an infectious disease model to real data.

Suggested book: "Modeling Infectious Diseases in Humans and Animals" by Keeling and Rohani.

Prerequisites: The only strict requirement here is calculus. Exposure to probability and programming experience will be a plus, but are not strictly necessary. There are multiple possible project directions here depending on student interest and experience.

Andrea Roberson

Mathematical Statistician at the U.S. Census Bureau

Description: Whether you desire to work at an airline that needs to improve customer retention, a financial services company that needs to mitigate risk, or a retail company interested in predicting customer purchasing behavior, your organization is tasked with preparing, managing, and gleaning insights from large volumes of data without wasting critical resources.

Traditional CPU-driven data science workflows can be cumbersome, but with the power of GPUs, your teams can make sense of data quickly to drive business decisions. In this project, you will learn how to build and execute end-to-end GPU-accelerated data science workflows that make it possible for you to quickly explore, iterate, and get your work into production. Using the RAPIDS™-accelerated data science libraries, you'll see how mathematics factors into a variety of GPU-accelerated machine learning algorithms such as XGBoost, DBSCAN, and logistic regression. We'll conclude by applying your new GPU-accelerated data manipulation and analysis skills to population-scale data to help stave off a simulated epidemic affecting the entire UK population.

Suggested Reading: Intro to Machine Learning, https://www.kaggle.com/learn/intro-to-machine-learning

Prerequisites: Accessible for all skill levels.

Malena Sabaté Landman

Postdoc in Applied Math at Emory University

Description: CT is an imaging technique that is used to reconstruct 3D objects from X-rays (we usually call this an inverse problem). Since we're not directly measuring the 3D object that we want to reconstruct, we need to use a mathematical model to recover a good approximation of the object of interest. In many clinical applications, we want to limit the number of X-rays taken since X-rays expose patients to radiation. However, having only a few measurements (aka sparse-data) makes the reconstruction process very challenging.

In this project, we will cover some basic theory and computational methods for sparse-data CT. After formulating this problem as a linear system, we will look at some of its properties (for example: why is this hard to solve?). We'll also study some of the algorithms that are currently used to solve this in real-world applications. To understand how this works, we will first simulate small toy examples and build up to reconstructing a brain (and, surprisingly, cheese) using real data.

A fun introduction to CT can be found in https://www.youtube.com/watch?v=dn358iX_WxQ (it's a long video, but for a snapchat you can watch from minutes 1:50 to 4:25!).

Suggested book: "Linear and nonlinear inverse problems with practical applications" by Jennifer L. Müller and Samuli Siltanen (2012). Other related reading we might consider are the interactive notes from Prof. Samuli Siltanen (these can be found in http://www.siltanen-research.net/IPexamples/) and the book "Discrete Inverse Problems: Insight and Algorithms" by Per Christian Hansen (2010).

Prerequisites: Basic linear algebra is needed, and some programming experience will be useful for a more hands-on project.

Meet the Students

Rasheed Jeheeb

Josh Kyei

Kobe Lawson-Chavanu

Morgan Lee

Naomi Logan

Nikira Walter