## The University of Pennsylvania Chapter

### What is the DRP?

The Directed Reading Program (DRP) is a program which pairs undergraduate students with graduate students for a semester-long independent study. It was started at the University of Chicago but now runs in mathematics departments all over the country.

There are no restrictions on choice of topics, and pairs will be assigned based on the interests of both undergraduate and graduate students.

For more information, feel free to contact the organizers Thomas Brazelton and George Wang, or our faculty organizer, Mona Merling.

### What is expected of mentees and mentors?

The mentors are expected to meet with their undergraduate mentees for an hour every week. In addition to this, the undergraduates are expected to work independently for a few hours every week and prepare for the meetings with their mentors. The mentors are also supposed to help their mentees prepare their talks for the final presentation session-this includes helping them choose a topic, go over talk notes and practice the talk.

Undergraduate participants will be registered for a pass/fail half-credit from the mathematics department for the DRP, so that the independent study can be reflected on their transcript.

### Presentations

At the end of the semester there will be a presentation session, which is open to all members of the departments and friends of the speakers.

## Apply

Spring 2021 pairings have already been made, please check back at the start of the Fall 2021 semester to apply.

# Spring 2021 Projects

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.

# Fall 2020 Projects

Asset Pricing Models
Zöe Patterson
Julian Gould
We will present a variety of classical asset pricing models, including CAPM, Arbitrage Pricing, and Risk Neutral Pricing. A short discussion of option pricing and stochastic processes will be included if time permits.
Category Theory with a View Toward Algebraic Topology (slides)
Andres Mejia
The goal of this project is to understand how a categorical viewpoint can shed light on various aspects of algebraic topology. We are trying especially to distinguish the "formal" categorical arguments from those of a fundamentally topological nature. We are working through Tom Leinster's "Basic Category Theory" and using some of these ideas to revisit the beginning chapters of Hatcher's Algebraic Topology.
Causal Inference and Machine Learning
Sam Rosenberg
Darrick Lee
It is often said that "correlation does not imply causation". This naturally raises the question, what evidence is sufficient to make claims about causation? Causal inference is the field in which such questions are answered. After introducing the necessary background of Structural Causal Models, we explore how tasks in machine learning like semi-supervised learning are cast in a new light when looking through the lens of causal inference.
Discrete and Continuous Dynamical Systems
Mark Lovett
Hunter Stufflebeam
Our goal for this project is to examine the differences and connections for discrete and continuous dynamical systems. This might include examining rules that work for both types of dynamical systems or examining different methods for solving each type of dynamical system.
Elliptic Curves and Modular Forms
Santiago Velazquez Iannuzzel
Zhaodong Cai
Elliptic curves feature in modern mathematics prominently; from solving Fermat's last theorem to cryptography, they are truly the globetrotter in the vast landscape of mathematics. Following Koblitz's book Introduction to elliptic curves and modular forms, in this project, we explore the fundamentals of the subject, first treating elliptic curves as one dimensional complex manifolds, then considering them defined over finite fields, finally studying modular forms as functions and differential forms defined on modular curves.
Fun with the Fundamental Group (slides)
Yi Ling Yu
Elijah Gunther
In this project we're learning about the fundamental group of a topological space starting from the definition of a homotopy. We'll then use basic properties of the fundamental group to work out basic topology problems. Depending on how far we get we'll learn about either the relationship between the fundamental group and covering spaces or learn some basic category theory and see how the fundamental group defines a functor.
Ideas in Non-Euclidean Geometry
Sarah Hayward
Ming Jing
We use Michael Henle's Modern Geometries to get the flavors of several types of Non-Euclidean geometry following the philosophy of the Erlanger Programm.
Knot Theory (slides)
Pedro Sacramento de Oliveira
Artur Bicalho Saturnino
We will study classical knot invariants and knot polynomials following Murasugi's book "Knot Theory and its applications". After we have laid this foundations will study some algorithms used in knot theory.
$p$-Adic Numbers and the Hasse-Minkowski Theorem (slides)
Nicholas Pilotti
Souparna Purohit
We will learn about some of the basic algebraic and analytic properties of the p-adic numbers, leading up to a proof of the Hasse-Minkowski theorem for quadratic forms over Q.
Random Walks (slides)
Ernest Ng
Eric Goodman
In this project, we will first study random walks on finite networks, then consider random walks on infinite networks, with the aim to understand Pólya's Recurrence Theorem. Using Doyle & Snell's "Random Walks & Electric Networks", we examine the applications of electric network theory to random walks. We will then look at related topics as time permits.
Set Theory
Patrik Farkas
Anschel Schaffer-Cohen
We will be carefully reading through Halmos's Naive Set Theory, which describes ZFC set theory via its axioms and their consequences.
Symplectic Geometry and Classical Mechanics (slides)
Sam Goldstein
Chris Bailey
In this project, we analyze the Lagrangian and Hamiltonian formalisms of classical mechanics in the language of differential geometry using symplectic manifolds. This allows us to view classical mechanics in terms of symmetries and their relationships with conservation laws. After developing the theory, we apply our results to analyze a variety of physical systems.
Topics in Stochastic Processes
Nicole Bobovich
Qingyun Zeng
The purpose of this project is to explore elementary stochastic processes and their effects on a variety of discrete random variables. We study the changes to the probability distribution functions of these random variables along with specific stochastic processes such as Markov processes, emphasizing the Chapman-Kolmogorov Equation and matrix representations of Markov Chains.
Traffic Flow Modeling and Car Accident Risk (slides and code)
Mihal Zelenin
Yao-Rui Yeo
Macroscopic modeling of traffic uses systems of differential equations to simulate the directed flow of cars on the road. However, most modeling of car accident risks is based on statistical analysis of historical crash data, rather than simulations of motion. Using flow equations and simulations to predict times and locations where car accidents are more likely to occur is a new area that is just starting to be explored in the past few years. In this project, I will look at the most widely used flow models for traffic, as well as the developing models that simulate car accidents. By running some of these models with different initial conditions, I will investigate factors that macroscopic simulations predict to increase risk, and check if they match historical data when such data is available.

# Spring 2020 Projects

Applied Topology (slides)
Scott le Roux
Aline Zanardini
The goal of this project is to learn how some tools and concepts from Topology can be used in the study of computational problems like Molecular Modelling, Data Analysis and Image Processing. We will first equip ourselves with the necessary background for understanding the required concepts from Topology and Algebraic Topology. Then look at how they can be applied to problems within the aforementioned fields.
Category Theory and Algebraic Topology (slides)
Abby Timmel
Thomas Brazelton
Category theory provides an abstract treatment of concepts that recur across many branches of mathematics. This project will focus on the application of category theory to algebraic topology. We will study some foundational concepts of category theory using Emily Riehl's "Category Theory in Context" and develop an understanding of algebraic topology through Allen Hatcher's "Algebraic Topology".
Classification of Coverings (slides)
Marielle Ong
For certain topological spaces X,  there is a bijection between the  isomorphism classes of its covering spaces  and subgroups of its fundamental group. Using Hatcher's "Algebraic Topology," we will learn about the fundamental group and look at several examples of how to calculate it. We will then explore the concept of covering spaces, universal covers, and how they relate to the fundamental group of the base space in order to understand this bijection.
Coding Theory
Valerio Galanti
Man Cheung Tsui
Over the course of the semester, we will study error-correcting codes in the context of information theory. After investigating Galois and field theory, we will focus on codes over algebraic curves. We plan to follow Serguei Stepanov’s book “Codes on Algebraic Curves,” supplemented by Hungerford as well as Dummit & Foote’s books on Abstract Algebra.
Cryptosystems (slides)
G. Esther Guan
Yao-Rui Yeo
The purpose of this project is to explore various cryptosystems and the mathematics behind them. We study classical cryptosystems such as RSA and the Diffie-Hellman key exchange in the first half of the semester, before focusing on elliptic curve cryptography for the second half. The main reference used is Neal Koblitz's "A Course in Number Theory and Cryptography".
Degree and Intersection Theory (slides)
Airika Yee
Artur Saturnino
The degree is a simple and powerful invariant of a differentiable map. We will see how this invariant and a generalization of it called the intersection number can be used to show some fixed point theorems such as the Poincare-Hopf, Borsuk-Ulam, and Lefschetz theorems as well as some other important theorems such as the Hopf and Jordan-Brouwer theorems. We will follow Milnor's "Topology from the Differentiable Viewpoint" and Guillemin and Pollack's "Differential Topology."
Diffie-Hellman Key Exchange and RSA Cryptosystem (slides)
Lisette del Pino
Tao Song
A fascinating mathematical fact underlying modern cryptosystems is that it is easy to multiply a positive integer times itself modulo a prime number, but it is very difficult to tell how many times it was multiplied by itself knowing only the integer and the result modulus prime. In this project, we use this fundamental fact, present an overview of the Chinese Remainder Theorem, and discuss Fermat’s Little Theorem in order to provide a basis for how the Diffie-Hellman public key exchange operates. We also investigate the RSA cryptosystem and Euler’s formula. Reference: "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher, and Silverman.
Lie Groups, Lie Algebras, and Representations (slides)
Dennica Mitev
Qingyun Zeng
In this project, I will introduce the key topics leading up to the motivation for Representation Theory and its significance, including Lie groups, their respective Lie algebras, and the homomorphisms between them. Using matrix Lie groups as a foundation, I will discuss how to construct such representations into a vector space and observe how their properties lead into an understanding of semisimple theory.
Parametric Time Series Analysis (slides)
Samuel Rosenberg
Darrick Lee
The purpose of this project is to give a brief survey of parametric time series analysis. The "big picture" ideas in this field will be discussed, with the white noise, random walk, and ARMA(p,q) models used as examples.
Probability Theory / Stochastic Processes
Lisa Zhao
Huy Mai
We will be following Durrett's Probability: Theory and Examples on topics including Central Limit Theorems, Martingales, and Markov Chains.
Topics in Causal Inference (slides)
Omkar A. Katta
The Rubin Causal Model inspired methods to answer questions about why real-world phenomena occur, including instrumental variables, regression discontinuity design, and differences-in-differences. In this presentation, we first briefly introduce the Rubin causal model. Then, using literature from statistics, economics, and computer science, we study richer variants of this static model in dynamic settings that capture interference effects, state dependence, and more. The resulting methods face such common issues as endogeneity and serial autocorrelation as well as new issues like quantifying uncertainty in the age of Big Data. Our goal is to obtain a broad overview of the interdisciplinary frontier of causal inference.
Principal Component Analysis  in a Linear Algebraic View (slides)
Anna Orosz
Jakob Hansen
In this project we will study Principal Component Analysis as a means for transformation of data. We will see how the best ellipsoid can be fit on the given data as well as the linear algebraic method to compute the PCA. We will go through the determination of the underlying components and reducing the number of components. It is also important to underline the theory behind the Singular Value Decomposition (SVD), specifically, how to compute the SVD and what we can expect as the result. Finally, we will go into detail about how to use SVD to use PCA and what the benefits and disadvantages of directly computing PCA without the use of SVD.

# Fall 2019 Projects

Analytic Number Theory
Suraj Chandran
Zhaodong Cai
It is said that Dirichlet's theorem on the infinitude of primes in arithmetic progressions invented analytic number theory. In this project, we will go over a proof of this theorem, which provides a good excuse to study Fourier analysis. Time permitting, we will also explore how analytic methods can be applied in other ostensibly discrete problems, for example prime number theorem and Lagrange's four square theorem.
Chromatic Symmetric and Quasisymmetric Functions
Zach Sekaran
George Wang
We will study symmetric function theory and its applications to graph colorings.  Every graph has an associated chromatic symmetric function, which is the generating function of the graph's proper colorings. Much of the work on chromatic symmetric functions is motivated by a long standing open problem that concerns graphs constructed from partially ordered sets and when the resulting chromatic symmetric function has positive coefficients in a certain basis. We will first follow Stanley's Enumerative Combinatorics Vol. 2 before reading Shareshian and Wachs' 2014 paper "Chromatic quasisymmetric functions".
Classical Mechanics
Annie Freeman
Benedict Morrissey
We will look at Lagrangian and Hamiltonian formalisms for classical mechanics following the books of Landau and Lifshitz, and Arnold.
Complexity Theory
Olivia Cheng
Jongwon Kim
We plan to study complexity theory and its intersection with mathematics. After learning the basics of complexity theory, we will focus on more specific problems of interest.
Elliptic Curve Cryptography
Carolina Mora
Souparna Purohit
The purpose of this project is to explore the basics of elliptic curves (focusing on the theory over finite fields) and to ultimately study their applications to cryptography (an approach known as Elliptic-curve cryptography). The main references are portions of Silverman-Tate's "Rational Points on Elliptic Curves", and portions of Silverman's "An Introduction to Mathematical Cryptography", among other scattered resources.
Elliptic Functions and Modular Forms
Ben Foster
Man Cheung Tsui
After briefly looking at how to analytically continue the zeta function and proving the prime number theorem, we investigate doubly periodic functions on the complex plane, modular forms, and modular curves. Our goal is to use the resulting theory to get formulas computing the number of ways a positive integer can be written as a sum of k squares. References: Stein and Shakarchi, Complex Analysis, Chapters 6, 7, 9; Diamond and Shurman, A First Course in Modular Forms, Chapters 1, 2, 3, (4).
Group Theory and Applications
Stephanie Wu
Thomas Brazelton
We will discuss group theory from basic principles, including set theory, subgroups, and quotient groups. After this we will discuss particular interesting examples, which may include groups of matrices and their applications in linear algebra, the braid group, and the fundamental group of a topological space.
Information Theory with Applications to Machine Learning
Chetan Parthiban
Jakob Hansen
Over the course of the semester, we hope to study some introductory topics in the field of information theory as well as their applications. This study will begin with a survey of some basic elements of the field such as entropy, data compression, and noisy-channel coding. We will spend the rest of the semester examining applications of information theory to statistical inference, with the goal being to reinterpret neural networks from an information theoretic perspective. We plan to follow David MacKay’s book "Information Theory, Inference, and Learning Algorithms," supplemented by Shannon's landmark paper "A Mathematical Theory of Communication."
Lie Groups and Control Theory
Anthony Morales
Darrick Lee
We will focus on Lie groups and Lie algebras with the motivation of exploring the intersection of abstract algebra and differential geometry. In addition, we will discuss its general applications to control theory, particularly regarding bilinear systems.
Quantum Theory and Representations of Groups
Mark Dsouza
Michail Gerapetritis
Our goal will be to understand 1-dimensional quantum systems, such as the quantum free particle, the hydrogen atom, and the harmonic oscillator, as well as develop some of the foundations needed for higher dimensional quantum field theories. The approach will focus on developing the necessary mathematical foundations on representation theory and analysis. Main reference: Peter Woit - Quantum Theory, Groups and representations: An Introduction.
Topology from the Differentiable Viewpoint
Airika Yee
Artur B. Saturnino
The degree of a smooth map is a powerful concept that is central to differential topology. In this project we will define this invariant and study how it is used to show the Fundamental Theorem of Algebra, Browder's Fixed Point Theorem, the Borsuk-Ulam theorem and, if we have time, the Poincare-Hopf and Hopf's Theorem. This project is based on Milnor's book with the same name.