Directed Reading Program

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

Fall 2020 pairings have already been made, please check back at the start of the Spring 2021 semester to apply.


Spring 2020 Projects

Applied Topology (Slides)
Aline Zanardini
Scott le Roux
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)
Thomas Brazelton
Abby Timmel
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
Adam Zheleznyak
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
Man Cheung Tsui
Valerio Galanti
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)
Yao-Rui Yeo
G. Esther Guan
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)
Artur Saturnino
Airika Yee
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)
Tao Song
Lisette del Pino
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)
Qingyun Zeng
Dennica Mitev
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)
Darrick Lee
Samuel Rosenberg
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
Huy Mai
Lisa Zhao
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)
Hadi Elzayn
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)
Jakob Hansen
Anna Orosz
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
Zhaodong Cai
Suraj Chandran
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
George Wang
Zach Sekaran
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
Benedict Morrissey
Annie Freeman
We will look at Lagrangian and Hamiltonian formalisms for classical mechanics following the books of Landau and Lifshitz, and Arnold.
Complexity theory
Jongwon Kim
Olivia Cheng
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
Souparna Purohit
Carolina Mora
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
Man Cheung Tsui
Ben Foster
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
Thomas Brazelton
Stephanie Wu
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
Jakob Hansen
Chetan Parthiban
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
Darrick Lee
Anthony Morales
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
Michail Gerapetritis
Mark Dsouza
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
Artur B. Saturnino
Airika Yee
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.