A Crash Course on Homotopy Theory

Michael Zeng

Andres Mejia

In this project we first study fibrations, cofibrations, homology, and related results. Then we will move on to big results in homotopy theory including the Whitehead theorem, the Hurewicz theorem, and obstruction theory.

The Continuum Hypothesis for Subsets of $\mathbb{R}$

Jacob Glenn

Julian Gould

The Continuum Hypothesis is an important historic conjecture in Set Theory regarding the cardinalities of subsets of the real line. While Gödel and Cohen showed that the continuum hypothesis is independent of the standard axioms of set theory, interesting questions about the continuum hypothesis remain. Are there classes of subsets for which the continuum hypothesis holds regardless of the truth of CH in general? By analyzing perfect sets we may resolve the continuum hypothesis for open and closed sets. Finally, we will discuss the continuum hypothesis for Borel sets and Analytic sets.

Differential Geometry

Patrik Farkas

Elijah Gunther

We're reading through Lee's Introduction to Smooth Manifolds, starting from the definition of a manifold. After we cover the foundational topics (manifolds, smooth functions, tangent spaces) we'll pick some of the more advanced topic(s) to read about, possibly Lie groups, flows along vector fields, or whatever other topic(s) look especially interesting depending on time.

Diophantine Equation and cryptography

Xuxi Ding

Andrew Kwon

We started from the Pell's equation and its application in cryptography. We also plan to explore some Elliptic Curve Cryptography.

DNA Topology

Elena Isasi Theus

Jacob van Hook

Our objective in this project is to gain a comprehensive and thorough understanding of the field of DNA topology. We will begin the semester by covering the basics of knot theory and algebraic topology, particularly the fundamental group. We will later progress through the topics contained in Bates and Maxwell's "DNA Topology", including DNA supercoiling, DNA on surfaces, knots and catenanes. After developing this theoretical basis, we will analyze its consequences in a biological setting, focusing especially in the role of topoisomerase proteins. Aside from this book, we plan to make use of "The Knot Book" by Colin C. Adams, the "DNA Topology Review" by Garrett Jones and Candice Reneé Price, and "Algebraic Topology" by Allen Hatcher.

Ergodicity Economics

Zöe Patterson

Hunter Stufflebeam

Motivated by newly published theories calling into question the ergodicity of economics, we plan to begin by understanding ergodic theory and its foundations such as measure theory and abstract algebra to uncover its implications in economics and finance. We will read "A Simple Introduction to Ergodic Theory" by Karma Dajani and Sjoerd Dirksin.

Game Theory and Competitions in the Bible

Angelica Sinay

Zoe Cooperband

This semester, we are planning to go over introductory concepts in the field of game theory, following the textbook "For All Practical Purposes" such as pure and mixed strategies of two-person conflict games, partial conflict games, larger games, and matrix representations of different competitions. In particular, we are going to analyze games within the Old and New Testaments of the Catholic Bible between God, saints, and humans. For the Biblical Games, we are following the paper written by Steven Brams called "Biblical Games: A Strategic Analysis of Stories in the Old Testament".

The Mathematics of Data Science

Sam Rosenberg

Darrick Lee

Using the preprint of the book Mathematics of Data Science by Bandeira, Singer, and Strohmer, we will explore some of the probability, linear algebra, and geometry behind commonly used techniques and algorithms in contemporary Data Science. Topics will include Community Detection via the Stochastic Block Model, as well as Linear Dimension Reduction with Random Projections using the Johnson-Lindenstrauss Lemma.

Primes in Arithmetic Progressions

Nick Pilotti

Ming Jing

We aim to prove Dirchlet’s theorem with the book Multiplicative Number Theory by Davenport. We will move into further topics in the book such as prime distributions if times allows.

Quadratic Forms and the Conway-Schneeberger Fifteen Theorem

Alex Kalbach

Yao-Rui Yeo

Over the course of the semester, we plan to work towards understanding the proof of the Fifteen Theorem. We will begin by acquiring foundational knowledge about $p$-adic numbers, followed by quadratic reciprocity. The beginning portion of Neal Koblitz's "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" will help guide us through this information. We will also investigate and learn how to apply two important statements, Hansel's Lemma and the Hasse-Minkowski Theorem. After this, we will begin to understand the idea of quadratic forms, as well as important theorems related to quadratic forms. From there, we will finally examine the proof of the Fifteen Theorem and its consequences.

Stochastic Processes

Abhi Bhandari

Leonardo Ferreira Guilhoto

Over the course of the semester, we aim to study the foundations of stochastic processes through a non measure theoretic lens, beginning with Markov chains before moving onto the theory behind martingales and renewal processes. We may also look into stochastic integration and Brownian motion. We plan to follow Gregory Lawler's "Introduction to Stochastic Processes."

$U(1)$ Gauge Theory

Oderico-Benjamin Buran

Jia-Choon Lee

Over the course of the semester we will study geometry using Morita’s “Geometry of Differential Forms” and Tu’s “An Introduction to Manifolds” to develop the underlying geometry of Gauge Theory. This will include learning about manifolds, differential forms, bundles, connections, etc. Then, we will use notes on lectures by Karen Uhlenbeck to formulate Maxwell’s equations as a $U(1)$ Gauge Theory.

Visualizations in Algebraic Geometry

Dennica Mitev

Jianing Yang

This semester we’ll begin with an exploration of the foundations of algebraic geometry in William Fulton’s book “Algebraic Curves”, covering topics such as Hilbert’s Nullstellensatz, affine varieties, local rings, and properties of plane curves. Afterwards, we’ll take a look at other varieties and morphisms as induced by the Zariski topology, with the goal of then investigating the visualizations that arise as a result of singularities and other blown up points.