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 Marielle Ong, 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 (bear in mind that Penn rules prohibit P/F classes from counting towards major/minor requirements).
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
Applications for Spring 2023 are closed and pairings have been made. If you'd like to be notified when Fall 2023 applications open, please send us an email at math-drp@sas.upenn.eduSpring 2023 Projects
A Variety-Focused View of Algebraic Geometry
Ethan Soloway
Avik Chakravarty
We plan to study some fundamental topics in algebraic geometry with a particular emphasis on varieties. We are beginning by studying affine varieties, their inherent connections to both Algebra and Geometry via the Nullstellensatz, and the Zariski Topology defined on them. From there we will spend some time studying the sheaf of regular functions and morphisms between varieties, before defining varieties generally separate from an affine context. Time permitting, we plan to study specific varieties including projective varieties and Grassmanians. Additional topics include birational maps and an introduction to schemes. The fundamental goal is to understand varieties as objects and the type of morphisms relating them before diving into motivated examples and specific uses.
We plan to use Andreas Gathmann's Notes on Algebraic Geometry as a guide, with other textbooks such as Algebraic Geometry by Hartshorne and Commutative Algebra by Atiyah and Macdonald as supplementary reading.
We plan to use Andreas Gathmann's Notes on Algebraic Geometry as a guide, with other textbooks such as Algebraic Geometry by Hartshorne and Commutative Algebra by Atiyah and Macdonald as supplementary reading.
A1 Milnor Numbers
Zhong Zhang
Thomas Brazelton
This semester, we will start with the classical story of the Milnor number and its role in understanding singularities of complex hypersurfaces. While the classical Milnor number at a singularity of a complex hypersurface could be expressed as the topological degree of the gradient of the defining polynomial, there is a similar story in the world of A1-homotopy theory: the Milnor form of a polynomial defined on an affine space at a singularity point is exactly given by the local motivic degree of the gradient of this polynomial. We will walk through these theories carefully and work on some fun problems if time allows.
Algebraic Topology
Angela Cai
Marielle Ong
We're planning on exploring introductory topics in algebraic topology by reading Algebraic Topology by Allen Hatcher. We begin by familiarizing ourselves with basic concepts such as homotopy, fundamental group, covering spaces, and etc. By the end of the semester, we hope to be able to use these tools to find prove algebraic invariants that classify topological spaces. Besides following Hatcher's book, we'll supplement the textbook reading with Aaron Landesman's "Notes on the Fundamental Group" and Laurentiu Maxim's "Algebraic Topology: A Comprehensive Introduction."
Algebraic Topology and the Mathematical Basis of Topological Data Analysis
Elena Isasi Theus
Maxine Calle
Over the course of the semester, we hope to study the mathematical underpinnings of TDA by studying algebraic topology. We will use a more theoretical approach than last semester, looking at singular homology, CW complexes, homological algebra, cohomology, and Poincare duality. On the more applied side we will go over some specialized ideas in persistent homology. We plan to follow a combination of “Algebraic Topology” by Hatcher and MIT's Algebraic Topology I notes by Prof. Haynes Miller. The applied ideas will mostly come from Dr. Vidit Nanda's notes on Computational Applied Topology
Causal Inference and Its Applications
Chenxi Leng
Miaoqing Yu
We hope to study the field of causal inference over the semester. We will start with the two main approaches in causal inference, the potential outcome model and causal graph, which are used for estimation and identification, respectively. After studying the foundations, we will dive into methods for solving those two problems. We also aim to study at least one up-to-date topic, like machine learning in casual inference, or its real-world applications in tech companies. We plan to use Brady Neal’s lecture notes "Introduction to Causal Inference from a Machine Learning Perspective" as a reference, supplemented by some selected research papers.
Chaos, Fractals, Dynamics and their Real Life Applications
Tise Ogunmesa
Yi Wang
The project will focus mainly on providing a summary of chaos theory, dynamics, and fractals. Then, real life examples ranging from fun and silly to significant and consequential will be touched upon. Thus, the project will focus on Chapters 1, 2, 5, 9, and 11 (subject to change) of the textbook: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz.
Cohomology of Sheaves
Eric Myzelev
Marc Muhleisen
A cohomology theory is a way of functorially assigning algebraic invariants to spaces, which among other uses can help identify when two spaces fail to be isomorphic. In this project, we will study examples of cohomology groups, give applications to enumerative geometry, discuss a relationship between cohomology and line bundles, and/or prove a recognition theorem for affine schemes. We will start with the basics of commutative algebra and category theory by going through some of "Commutative Algebra" by Atiyah and MacDonald and "Category Theory in Context" by Emily Riehl. We will then move to "Algebraic Geometry" by Hartshorne.
Computation and Complexity with Category Theory
Ruxandra Icleanu
Julian Gould
We plan to understand how to formalise concepts from theoretical computer science using category theory, with Yanofsky's "Theoretical Computer Science for the Working Category Theorist". We start by learning how to translate different models of computation in categorical definitions, and how these models are linked together. We then hope to dive into computability and complexity theory.
Concepts and Theories of Ricci Flow and its Consequences
Ling Xu
Jacob Van Hook
Throughout the semester, we will study Hopper's book "The Ricci Flow in Riemannian Geometry". We will first cover the basics of Ricci flow, including the background material in Riemannian geometry, and then explore the short-time existence and uniqueness of solutions to the flow equation, as well as the stability of solutions under perturbations. In the rest of the semester, we hope to investigate the geometric and topological consequences of the Ricci flow, including the behavior of the curvature tensor, the formation of singularities, and the implications for the topology of the manifold. If we have time, we also hope to discuss the relationship between Ricci Flow and other curvature flows such as mean curvature flow.
Deep Learning
Boya Zeng
Leonardo Ferreira Guilhoto
This semester, we plan to focus primarily on the textbook "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville. We will closely read Part II (Modern Practical Deep Networks) to gain a deeper understanding of the structure, mechanisms, and mathematical foundations of modern neural networks. After finishing Part III (Deep Learning Research) of the book, we will select a specific topic of interest to explore in depth. Tentative topics include large-scale transformer-based language models and text-to-image diffusion models.
Exploring Algebraic Geometry
Max Wang
Deependra Singh
Throughout the semester, I will be reading Algebraic Curves by William Fulton, with the goal of exploring the Riemann-Roch Theorem.
Formal Methods of Verification in the Context of Type Theory
Kaan Erdogmus
Oualid Merzouga
Over the course of this semester, we aim to study some topics in Type Theory with a focus on applications in Formal Verification. Formal Verification languages like "Coq" have been used for writing the machine-checked proofs of major theorems in various areas of Math, as well as for formally proving the correctness of software, in particular, for mission-critical systems. The research therefore also overlaps with proof theory. We have started off the semester with reading the material for coursework in the Computer Science departments, in particular with CIS 341 and 500 (Compilers and Software Foundations) that touch on Type Theory and provable correctness, and will be following up with relevant papers once the exact topic is further specified.
Foundations of Differential Geometry and Topology
Eric Yu
Benjamin Keigwin
In this project, we will cover the foundational material in point-set topology to begin talking about smooth manifolds. Roughly aiming to work through the first few chapters and Appendix A of John M. Lee's "Introduction to Smooth Manifolds," we intend to cover some of the theorems traditionally stated in calculus, and see them in the more general setting of smooth manifolds. The principal aim is to see how one can treat spaces that are locally familiar, i.e. locally Euclidean, in a way similar to the way one thinks about multivariable calculus.
Harmonic Analysis and Transforms
Ekaterina Skorniakova
Travis Leadbetter
This semester, we will be diving into a study of harmonic analysis, transforms, and their applications in electrical engineering and robotics. We will start with reading several chapters in "Applied Analysis" by Cornelius Lanczos, and then we will move onto a more involved text within a more specific topic.
Information Theory with Focus on Quantum Entanglement
Shuyi Wang
Christopher Bailey
Throughout this semester, we hope to study the building blocks of information theory and their specific uses in applications of quantum entanglement. We will start with basic concepts in information theory, such as entropy, statistical mechanics, and conservation of information. As I develop a solid understanding of this field, we will then dive into its interface with quantum mechanics, with the goal of investigating the transmission of mutual information between entangled particles. Finally, we will look into the applications of quantum entanglement in communications, computations, and radar systems.
Introduction to Model Theory
Elan Roth
Jin Wei
The primary reading for this semester is Model Theory: An Introduction by David Marker. We hope to read about the foundations of Model Theory as a mathematical topic rather than as the bedrock of mathematical logic. We also plan on searching deeper into the pure side of model theory rather than exploring any of its applications, allowing us to begin working on open questions in the field. Our goal is to work on problems specific to model theory within the realm of mathematical logic and aim to make progress in an open problem.
Linear Algebraic Groups
Santiago Velazquez Iannuzzelli
Yidi Wang
We are going to look at linear algebraic groups and look at their structure theory from Humphrey’s and Springer’s books.
Matroids & Combinatorial Optimization
Vibha Makam
Zoe Cooperband
Over the semester, we hope to delve into understanding more about matroids and how they are used in combinatorial optimization. We will start by reading introductory papers about matroids as well as combinatorial optimization to gain a better understanding. We are still exploring what main references to use for the project.
Modular Forms
Chenglu Wang
Souparna Purohit
According to Martin Eichler, there are five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. We will begin with studying some basics of L-functions, modular forms, and Hecke operators. Then, we will read some elliptic curves, and finally spend the rest of the semester on examining the connections between modular forms, elliptic curves, and number theory. We plan to follow Chapter 7 of Serre's "A Course In Arithmetic" and Silverman's "The Arithmetic of Elliptic Curves."
Optimal control theory and dynamic programming
Ayanav Roy
Hangjun He
Over the semester, we hope to learn the basics of optimal control theory and use dynamic programming to solve the problem. We will first begin with linear time optimal control and talk about the Pontryagin maximum principle. Then, we will learn dynamic programming - a widely used approach to solve both linear and nonlinear control. Some interesting applications in financial economics will be introduced if time permits.
Probability Theory
Evan Qiang
Jae Choi
The goal of this project is to get become familiar with the most important results in measure-theoretic probability theory such as the Law of Large Numbers and the Central Limit Theorem. We will follow Durrett's "Probability: Theory and Examples" and Heil's "Introduction to Real Analysis."
Reductions of Elliptic Curves in Finite Fields
Quincy Alston
Jianing Yang
Observe that Q is the field of fractions of Z. Consider an elliptic curve E defined over Q. One can transform E into a curve over Z by clearing denominators and using Weierstrass normal form to create an isomorphic elliptic curve over Z. If we endow Z with the Zariski topology then prime ideals of Z can parametrize the Weierstrass form of E together with the reductions of E in finite fields. Certain properties of E can be verified by confirming those properties across the reductions of finite fields. We plan to read on this method and its generalizations for other schemes and rings.
Riemannian Manifolds
Arjun Shah
Elijah Gunther
We are going through Lee’s book for Riemannian Manifolds and are going to eventually go more in depth into a topic as we progress. Riemannian manifolds and their relation to topology is one direction.
Understanding Generating Functions
Darren Zheng
Xinxuan (Jennifer) Zhang
The goal of this project is to see the widespread use of generating functions in counting particularly difficult sequences that may not have a closed form. In many cases, the generating functions captures a lot of information. We will begin the semester by studying general combinatorial principles and tools using "The Art of Counting" by Bruce Sagan. Afterwards, we will focus more on generating functions and specific examples of their use in combinatorics, such as rook polynomials. We seek to understand generating functions more rigorously and how to manipulate them. After a few chapters, we will switch to select sections of Enumerative Combinatorics, Vol. 1 by Richard Stanley.
Understanding Optimization Algorithms for Graph Alignment Problem
Yuntong Fu
Xinrui Yu
Graph alignment algorithms seek to find a correspondence between different topological structures. The study of such algorithms span multiple subjects and are of great interest to fields like computational geometry and computer vision.
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Lex Giglio
Xiangrui Luo
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Mike Zhou
Andrew Kwon
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Sophie Kadan
Christopher Bailey
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Zakaria Sines
Miguel Lopez
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Yiyang Liu
Tianyue Liu
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Fall 2022 Projects
Zhaobo (Tom) Han
Elijah Gunther
In this project our main goal is to gain some familiarity with some important topics and tools in algebraic geometry, such as projective and affine varieties, singularities, and the Riemann-Roch theorem. These topics will be studied from the perspective of algebraic curves. The main text for this project will be William Fulton's Algebraic Curves-An Introduction to Algebraic Geometry, and we will probably refer to Atiyah's or Eisenbud's textbooks if background knowledge from commutative algebra is needed.
Algebraic Geometry and A1-Homotopy Theory
Zhong Zhang
Thomas Brazelton
This semester, we hope to study algebraic geometry and A1-homotopy theory. We will start with basic algebraic geometry concepts like varieties, schemes, and sheaves. After laying the foundations, we will spend the rest of the semester learning about “homotopy invariants” in algebraic geometry. Hopefully, before the end of the semester, we will get to some fun topics like Serre's problem and Bass-Quillen conjecture. For reference, we will mainly follow Aravind Asok’s notes “Algebraic geometry from an A1-homotopic viewpoint".
Analysis of Curve Shortening Flow, Mean Curvature Flow, and the Applications.
Ling Xu
Hunter Stufflebeam
This semester we hope to build off of differential geometry concepts studied last semester in looking at curve shortening flows and the behaviors of specific curves such as convex curves and solutions to the evolution equation such as the Grim Reaper. We hope to eventually extend to higher dimensions with mean curvature flows and possibly Ricci flows using the Gage-Hamilton papers and Halshofer notes, complemented with Lee's "Riemannian Geometry".
Vikram Balasubramanian
Ziang Niu
This semester, I hope to get an introduction to causal inference as well as specific details about matching techniques. I had previously completed Wharton’s DRP on a similar topic, so I will start with a more mathematically, theory-based understanding of causal inference, focusing on Donald Rubin’s potential outcomes approach. We are following Rubin and Imbens’ book “Causal Inference for Statistics, Social, and Biomedical Sciences.”
Classical mechanics and symplectic geometry (slides)
Yiyang Liu
Tianyue Liu
We will first be working toward Noether's Theorem, and then take the classical mechanics as motivation for symplectic geometry. To achieve so, we plan to go through selected chapters of Arnold's Mathematical Methods of Classical Mechanics, and then switch to learning about symplectic geometry through lectures notes by Ana Cannas da Silva, which contains many homework problems as practice.
Computability Theory and the Turing Machine
Michelle (Tianchen) Gu
Krishan Canzius
This semester, I hope to gain a basic understanding of computability theory and the
Turing machine. Under the guidance of Krishan Canzius, I would explore areas like the
definition of computability, computably enumerable sets, and Turing reducibility. I will
use Robert Soare’s Turing Computability - Theory and Applications as the basis for my
study, and hope to go through its first three chapters.
Quincy Alston
Jianing Yang
The inverse Galois problem is open over the rational field. We are interested in what Galois groups are possible when you adjoin a point on a hyperelliptic curve to the rational field such that the extension is Galois. We will be using geometric methods by means of computation to recognize patterns in the hyperelliptic curves which realize non symmetric Galois groups and will be focusing on using the group law over elliptic curves to prove new results. We will be using Sage and Python to compute explicit examples and make conjectures which we will attempt to prove. This project has an emphasis on cyclic Galois groups and proving results for the Cyclotomic polynomials.
Iris Horng
Jacob Van Hook
Over the course of the semester, we hope to study concepts from classical Morse theory and discrete Morse Theory to see how they can be applied to simplicial complexes to analyze their topology. We will also learn about its applications to fields in applied mathematics and computer science, such as topological data analysis, which will also lead us to explore how persistent homology can be computed using discrete Morse Theory. We plan to follow and reference “Discrete Morse Theory” by Nicholas A. Scoville.
Fundamentals of Computational Applied Topology
Elena Isasi Theus
Maxine Elena Calle
Over the course of the semester, we hope to study the mathematical underpinnings of computational applied topology by studying algebraic topology and its applications. We will be starting with some topology and algebra preliminaries, as well as some ideas in algorithms. Afterwards, we have conceived this as a journey from the pure to the applied: we will be starting with a little bit of homotopy theory, then go deeper into homology and cohomology; later, we will look into the applications of these ideas to single cell data analysis and (as time permits) time series analysis, network and neuroscience. We plan to follow a combination of “Algebraic Topology” by Hatcher, “Computational Homology” by Kaczynski, Mischaikow and Mrozek, the “Computational Algebraic Topology” lecture notes by Vidit Nanda, and “Topological Data Analysis for Genomics and Evolution” by Raúl Rabadán and Andrew J.Blumberg.
Graph Learning from Signals
Aryan Singh
Xinrui Yu
Over the course of the semester, we will study several topics related to learning graphs from signals in data. We will begin with reading modern research on how to solve the graph learning problem from signals and techniques in classical signal processing. Then, we will specialize into topics involved in improving robustness and efficiency and enhancing model interpretability using signal processing ideals. Specifically, then we will narrow into the connections between GSP and machine learning, with a focus on probabilistic techniques. We will follow a variety of cutting edge papers on the topic.
Graph Theory and its Applications
Ekaterina Skorniakova
Travis Leadbetter
This semester, we are doing a deep dive into graph theory and studying its applications. We will mainly be reading from "Pearls in Graph Theory" by Nora Hartsfield and Gerhard Ringel, and we will be using "Graph Theory" by Reinhard Diestel as a complementary text. The goal is to understand the basics of graph theory, and gain appreciation for the subject by looking into its applications to algorithms, machine learning, circuitry, and robotics.
Sophie Kadan
Noah Braeger
In this project, we will introduce the notion of using Riemannian geometry to model gravity. We
will use tools such as the Riemann curvature tensor to quantify the curvature of spacetime, and explain the
preferred trajectories in which particles travel in curved spacetime. We will define the curvature tensor and
these trajectories with objects such as the covariant derivative. Finally, after motivating Einstein’s equation
as a fundamental relation between the curvature of spacetime and the stress-energy tensor, a perturbative
solution will be introduced and interpreted as gravitational waves
Introduction to Proof Theory
Santiago Velazquez
Alvaro Pintado
We plan to study classical proof theory. This will involve learning the well-known proof systems developed by Gentzen for Natural Deduction and the Sequent Calculus. A first end goal is to understand the Normalization and Cut Elimination results for these systems. A more ambitious goal is the consistency proof of Peano Arithmetic given by Gentzen. Along the way, we plan to understand the difference between intuitionistic logic and classical logic.
We plan to read from the book: Elements of Logical Reasoning - Jan von Plato.
Invariants in Algebraic Topology
Darren Zheng
Yi Wang
We hope to study some of the theory in the fields of topology and knot theory. In particular, the study of classifying and invariants used to distinguish spaces/knots. We begin the semester with a survey of standard techniques using Topology of Surfaces by L. Christine Kinsey and The Knot Book by Colin Adams. We will end it by reading the first chapters of Hyperbolic Knot Theory by Jessica Purcell.
Manifolds and Information Geometry
Sophie Abner
Marielle Ong
For this project we will be studying manifolds, metrics, curvature, and connections, eventually leading to a study of information geometry. We will be reading An Introduction to Manifolds by Loring W. Tu, and Introduction to Smooth Manifolds by John M. Lee.
NFT Auction Mechanism Design
Stephanie Wu
Jennifer Wang
In this project, we will discuss a framework for decentralized NFT auction mechanism
design. More specifically, we will define some formal rules associated with single-item NFT auction mecha-
nisms and consider some viable examples of such. Time permitting, we will examine some important design
challenges and possible ways to overcome these challenges
Joshua Anumolu
Leonardo Ferreira Guilhoto
Operator learning is a new field of machine learning that uses neural networks to learn maps
between infinite dimensional function spaces. In this project, we will cover the fundamentals of this
field and some of its recent developments. Starting with some definitions, we distinguish operator learning
from classical machine learning. We will then move on to describe a basic architecture and some examples
of applications.
Persistent Homology
James Blume
Miguel Lopez
Over the course of the semester, we hope to study persistent homology. The study will start with reading the textbook Computational Algebraic Topology. From there, we plan to read papers applying persistent homology to concepts like machine learning and natural learning processing.
Probabilistic Methods (video)
Max Wang
Deependra Singh
Planning on exploring probabilistic method on Ramsey number bounds, and other interesting problems.
Mike Zhou
Eben Blaisdell
A study on proofs and their connection to models. Following a combination of notes by Townser and Model Theory: An Introduction by Marker, we will explore different proof systems, their rules, and special properties (such as cut elimination). We will explore how completeness and models can simplify proofs within semantics.
Jack Blackman
Souparna Purohit
For my project I will be looking into Number Theory, more specifically the idea of quadratic reciprocity. I will mainly be using a paper from the University of Mississippi that explains the concepts background, as well as details a few of its proofs. I have found some other sources detailing other of Gauss' 8 proofs. Finally I plan to investigate some of the proof's applications such to concepts like Fermat's theory on the sum of two squares.
Ruxandra Icleanu
Julian Gould
We started with classical type theory, using Girard's "Proofs and types". We then moved to dependent type theory, with Rijke's "Introduction to Homotopy Type Theory." After this, we learned how to use the type theory-based proof assistant Coq and its library UniMath.
Spring 2022 Projects
Cianán Conefrey-Shinozaki
Maxine Calle
An n-dimensional topological quantum field theory is a symmetric monoidal functor from the category of n-dimensional cobordisms to the category of vector spaces. In essence this means we can draw n-dimensional manifolds with closed, oriented (n-1)-dimensional manifolds as boundaries, and associate them with state spaces and time-evolution operators in linear algebra. In physics this corresponds to the use of Feynman path integrals, which associate simple pictorial diagrams with the evolution of a quantum mechanical system through time. As a foundation for this project we will be reading Joachim Kock's Frobenius Algebras and 2D Topological Quantum Field Theories, and then supplementing this with other literature that explores the topic from a physics perspective.
Algebraic Topology: Fundamental Groups, and the Exploration of How Algebra Informs Tolopogy and the Converse
Angel Munoz
Yi Wang
Throughout the semester, we look forward to surveying tolopogy and how it may inform algebra. We began with point-set topology, and will delve into homology, homotopy and the idea of a fundamental group. We began with the first three chapters of James Munkres' 'Topology' and will pivot into following L. Christinie Kinsey's "Topology of Surfaces" for most of the section. Once that foundation is established, the plan is to supplement with Hatcher's "Algebraic Topology" and narrow in on the concept of a free group and applications of homology.
Mike Zhou
Andrew Kwon
A study on the distribution of primes. Following a combination of Analytic Number Theory by Apostol, Multiplicative Number Theory by Davenport, and Terence Tao’s notes, we will explore proofs of the Prime Number Theorem as well as the importance of zeta functions.
Axiom of Choice and its Consequences
Zakaria Sines
Krishan Canzius
We plan on reading through general topics of Set Theory, with a focus on the Axiom of Choice, its logically equivalent statements, and its consistency. Part of what we'll be reading will be the fundamentals of formal logic, how consistency of a set of axioms is proven, and the consequences of the axioms of Set Theory. A nice captsone for this project will hopefully be a presentation on the proof of the consistency of Choice. The main reference will probably be "Set Theory" by Kenneth Kunen.
Category Theory and Simplicial Sets
Carolina Mora
Marielle Ong
Establishing a basis in category theory, we will go through Perrone’s “Notes on Category Theory”, focusing on basic definitions, the Yoneda lemma, limits/colimits and adjunctions. We will then transition into simplicial sets, particularly as realized into geometric, topological structures, using texts by Friedman, Riehl and Goerss and Jardine.
Combinatorial Games and the Surreal Numbers
Richard Fried
Joanne Beckford
We will explore combinatorial games and the various methods for how to solve them. As discovered by John Horton Conway in the 1970's, this naturally leads to an entirely new number system called the surreal numbers that contain the reals, infinitesimals, and transfinite numbers. My presentation will introduce this system through the games that sparked its discovery then end with several open problems in this new system of numbers that are being worked on today.
Shriya Karam
Leonardo Ferreira Guilhoto
This semester, we will explore the mathematical theory behind stochastic queuing models as well as queuing networks. We will focus on queuing diagrams, Markov chain representations of queues, and steady state behavior of single and multiple server queues with finite and infinite capacity. We will follow William Stewart's textbook "Probability, Markov Chains, Queues, and Simulation" as well as supplemental papers on queueing theory in traffic flow modeling.
Differential Forms in Algebraic Topology
Zhaobo (Tom) Han
Andres Mejia
In this project, we will focus on examining differential forms from an algebraic-topological perspective. This subject lies at the intersection of several subjects, such as algebraic topology and differential geometry. The primary text we will use is the book with the same title written by Raoul Bott and Loring Tu. Over the course of the semester we will also use other texts (such as Foundations of Differentiable Manifolds and Lie Groups by Frank Warner) occasionally to facilite our understanding of the concepts/theorems mentioned in the text.
Differential Geometry and Applications of Mean Curvature Flow
Ling Xu
Hunter Stufflebeam
Over the course of the semester, we hope to study mean curvature flows and the interpretations in differential geometry, PDE's and more. We will start with Pressley's Elementary Differential Geometry to learn about properties of curves and spaces as a foundation to our exploration of curvature flows and how they turn complex geometric objects into nicer ones, starting with the simplest curve shortening flows. We will also analyze the applications involving curvature flows and topological concepts such as identifying spaces, and hopefully learn about minimal surfaces and their theories as well.
Zhong Zhang
Thomas Brazelton
This semester, we will continue reading Sheldon Katz’s “Enumerative Geometry and String Theory”. We hope to learn more tools for counting finitely many geometric structures, like rational curves on a quintic threefold. We will go over some theories in Physics like Mechanics and Quantum Field Theory. After building foundations, we will spend the rest of the semester investigating the interesting connection between String Theory and Enumerative Geometry, which is made clear by Topological Quantum Field Theory and Quantum Cohomology.
Exploring Algorithmic Trading in Prediction Markets
Justin Lipitz
Ryan Brill
This semester, we are exploring algorithmic trading and surveying various ideas across mathematics like Topological Data Analysis via the academic prediction market "Predictit.com." We will begin by obtaining data on the prediction market through a package in R prior to applying these mathematical techniques in our analysis. Right now, we aren't following a specific text but are currently utilizing various publications about algorithmic trading.
Fractals
Patrik Farkas
Elijah Gunther
This project will be an introduction into fractals along with a discussion of several examples like Sierpinski triangle and Cantor set. We will also discuss some measure theory as it underlies some fractals and their properties.
Quincy Alston
Eben Blaisdell
This semester, we are looking to get an overview of the field of mathematical logic and its formalization of logical reasoning. We are going to relate mathematical logic to number theory and explore what sets of numbers are computable and what sets can Gödel's theorem be applied. We plan to focus heavily on how Gödel's theorem qualifies the computability of a set of numbers and consequences in computer science applications. We are guided mainly by Christopher Leary's book "A Friendly Introduction to Mathematical Logic."
Gravitation and differential geometry (slides)
Yiyang Liu
Chris Bailey
For this directed reading project, we would possibly look into the connections between gravitation and differential geometry. The goal would be to try to understand gravitation while learning the required mathematics, especially the geometry. We use for reference the class notes on general relativity published by David Tong. In the end, if there is time, we might look into how gravitation relates to information theory.
Nick Pilotti
Souparna Purohit
We will learn about Lie groups, Lie algebras and their representations. Avoiding the theory of manifolds at first, we will use Brian Hall’s text to become familiar with the ideas of matrix Lie groups and their associated Lie algebras. Then, we will study the representations of Semisimple Lie Algebras. If time permits, we aim to understand the more general theory where Lie groups are defined as a manifold.
Persistent Homology Application to Data Science
Mason Larkin
Miguel Lopez
Persistent homology is a method used in topological data analysis (TDA) to study qualitative features of data that persist over different scales. In this project, we will be exploring topics in topology and their applications to data science and machine learning.
Probability Theory with Measure Theory
David Kogan
Jae Choi
I plan to follow "Probability: Theory and Example" by Rick Durrett. I will supplement this text by "Probability and Measure" by Patrick Billingsley.
Stochastic Differential Equations
Brian Lee
Artur Bicalho Saturnino
This semester, we will study stochastic differential equations and their applications in finance, biology, and physics using F. C. Klebaner’s “Introduction to Stochastic Calculus,” supplemented with L. C. Evans’ “Partial Differential Equations.” We will begin with a review of basic discrete-time stochastic processes, then spend the rest of the semester covering Brownian motion and martingales, Ito calculus, and the connection between stochastic processes and ODEs/PDEs.
Kevin You
Jin Wei
In this project, we will go over the ZFC axioms towards constructing regular mathematical objects. We argue that all regular mathematics can be done in an extremely small set that we define recursively. That is, the natural numbers, the rationals, the reals, and the complex numbers and all possible algebraic operations structures on these sets belong to this set.
Topological Data Analysis and Financial Math
Kevin Li
Julian Gould
Over the course of the semester, we hope to study some introductory topics in financial math. We also hope to study and understand topological data analysis, and potentially it's applications in markets.
Topological Data Analysis and Music
Iris Horng
Jacob Van Hook
Over the course of the semester, we hope to study concepts in topology like simplicial complexes, homotopy, and homology, which will allow us to look at persistent homology and topological data analysis. We will then explore topological structures in music and learn how topological data analysis can help analyze musical data. We plan to follow and reference “Computational Algebraic Topology Lecture Notes” by Vidit Nanda, supplemented by “Topology of Musical Data” by William Sethares and Ryan Budney.
Ekaterina Skorniakova
Travis Leadbetter
Inspired by GRASP/MODLAB's work on the Variable Topology Truss (VTT) robot, we plan on first covering introductory topics of topology, including set theory, connectedness, and compactness. We then plan on diving deeper into concentrated topics in geometric topology and configuration spaces. For the introductory part of the semester, we plan on reading "Principles of Topology," by Fred H. Croom. More focused resources for geometric topology and configurations spaces will be chosen depending on the time/resources available. We will also supplement the reading by watching Northwestern University's lectures on "Modern Robotics: Mechanics, Planning, and Control.
Type Theory and Formal Methods
Alexander Kassouni
Alvaro Pintado
We plan on studying the foundations of type theory, ultimately building up to the calculus of contstructions. We hope to get through the first 10 chapters of "Type Theory and Formal Proof" by Nederpelt and Geuvers, which describes each component of the lambda cube. We then plan on studying how the calculus of constructions is a basis for modern proof assistants, with some research into either Lean or Coq. If time allows, we also hope to study the correspondences between type systems, logic, and category theory.
Fall 2021 Projects
Robin Murugadoss
Ming Jing
The goal is to read the textbook “Algebraic Curves” by Fulton in order to gain an understanding of basic algebraic geometry techniques, one of the most prominent fields in Mathematics. The student is expected to finish 5 chapters of the textbook (or more as time permits), to begin reading about some of the applications of this area of mathematics towards the proofs of already-known geometrical theorems.
Applications of Game Theory in Non-Traditional Areas
Lea Cesaire
Julian Gould
We will study some applications of Game Theory in the Scattered food problem, in order to do that we will go over the paper "Behavioral Despair in the Talmud". We will primarily be interested in applications of game theory in religions,foreign affairs, and diplomatic relations between countries. We hope to later on take a look at "On Numbers and Games" by John H Conway.
Category Theory
Patrik Farkas
Elijah Gunther
General study of categories and functors, proving and studying Yoneda lemma, studying of limits and colimits and looking at examples of the categories
Kevin You
Deependra Singh
Over the course of the semester, we hope to study some basic topics in algebraic topology starting with covering spaces. The goal will be to look at algebraic structures once a basic understanding of the topological notions is developed, and to examine instructive examples. The books used will be Munkres' "Topology" and Hatcher's "Algebraic Topology".
Enumerative Geometry and String Theory
Zhong Zhang
Thomas Brazelton
Over the course of the semester, we hope to study some introductory topics in the field of enumerative geometry as well as its relationship with the String Theory in Physics. We will start with projective space, stable maps and manifolds, after which we will get into cellular decompositions and rational curves. Ideally, if we still have time, we will spend the rest of the semester studying its applications in physics, such as String Theory and the Topological Quantum Field Theory. We plan to follow Sheldon Katz’s book “Enumerative Geometry and String Theory”.
Introduction to Vector Bundles
Cianán Conefrey-Shinozaki
Sammy Sbiti
In this project, we will learn about vector bundles, study their motivation and develop an intuition for the geometry and topology of these structures on surfaces. We will gain an understanding of the theory by looking at examples such as the tangent bundle to the unit 2-sphere in R3. Such isomorphisms between line bundles on the unit 1-sphere, Möbius strip, and real projective line
Magical (Expander) Graphs in Math and Computer Science
Daniel Lee
Oualid Merzouga
We will be studying the theory and applications of expander graphs. To start, we'll cover how expansion can be defined in combinatoric, probabilistic, and algebraic language. We'll then explore how each of these formulations reveals different properties of expander graphs, exposing a wide range of applications; from MCMC sampling, to group theory, to the derandomization of randomized algorithms. In parallel, we'll study methods for actually constructing these "magical graphs". We plan to use "Expander Graphs and Their Applications," a survey by Hoory, Linial, and Wigderson, as a starting off point, using its (many) references as additional resources.
Iris Horng
Jacob van Hook
Over the course of the semester, we hope to study some introductory topics in groups and representations which will allow us to look at matrix lie groups. This study will begin with a better understanding of groups and fields in order to dive deeper into an approach for learning the basics of Lie group theory. We plan to follow Brian C Hall’s paper “An Elementary Introduction to Groups and Representations,” supplemented by Kristopher Tapp’s book “Matrix Groups for Undergraduates.”
Methods in Elliptic Curve Cryptography
Dennica Mitev
Jianing Yang
This semester, we studied modern methods in elliptic cryptography, many of which are improvements on classical methods due to their heightened security. In particular, we looked into the elliptic versions of the Diffie-Hellman Key Exchange and Elgamal Public Key Cryptosystem, Lenstra’s Elliptic Curve Factorization Algorithm, and Supersingular Isogeny Graphs, along with the appropriate contexts for each.
Nick Pilotti
Souparna Purohit
Modular forms have surprising connections to seemingly unrelated objects in mathematics, such as the Monster group and the equation behind Fermat's Last Theorem. This semester, we will begin to study modular forms along with their relationship to complex elliptic curves and modular curves. As an introduction to the subject, we will read Chapter 7 of Serre's A Course in Artithmetic. Then we will read the first few chapters of Diamond and Shurman's A First Course in Modular Forms, which aims to explain the Modularity Theorem, a result that states "All rational elliptic curves arise from modular forms."
Kahnrad Braxton
Chris Bailey
Initially we are reading through the first few chapters of 'Elementary Differential Geometry' by Andrew Pressley to acquire some background knowledge on Differential Geometry. Next, we will extensively study the chapter titled Hyperbolic Geometry to get some expertise, and then formulate our question from this study.
Simplicial Homotopy
Michael Zeng
Maxine Calle
Over the course of this project, we aim to get comfortable with simplicial homotopy theory and study its applications in e.g. category theory. We begin by reading introductory materials about simplicial sets and Kan complexes. Then, we go on to studying homotopy colimits, model categories, and classifying spaces. Some of the references are "an elementary illustrated introduction to simplicial sets" (Friedman), "Leisurely intro to simplicial sets" (Riehl), "A Primer on Homotopy Colimits" (Dugger), and "Simplicial Homotopy Theory" (Goerss & Jardine). (Sorry for the late submission!!)
Stochastic Processes
Maxmillian Tjauw
Léo Guilhoto
Over the course of this semester, we will read through "Introduction to Stochastic Processes" by Lawler. We have covered finite and countable Markov Chains, optimal stopping, and lastly, Martingales. In particular, we will go through a few examples of martingales and show how a martingale betting strategy can be used to win in fair games.
Symmetric Functions and Representation Theory (video)
Richard Fried
Xinxuan Wang
Our project begins with studying representation theory, giving particular emphasis to representations of the symmetric group. Then, we focus on Young Tableaux, the Robinson-Schensted-Knuth algorithm, and symmetric functions with the goal of understanding the directions of current research in this area. Our study of inquiry roughly follows the defining text The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions by Bruce E. Sagan.
Topics in Statistics and Data Science: Modern Optimization-based Methods in Causal Inference
Mason Larkin
Siyu Heng
"Through this directed reading program, we will study some selected topics in statistics and
data science. Specifically, we will focus on the applications of modern data science techniques
(in particular large-scale optimization) in causal inference (in particular observational studies).
We will learn how to use modern data science techniques and computational methods to rigorously draw causal
conclusions using observational data.
Topological Data Analysis (slides)
Joshua Ibrahim
Miguel Loopez
This semester will be an introduction to the use of topological tools in data
analysis and to combining these tools with methods from statistics and machine learning. Persistent homology was introduced by Carlsson about 20 years ago, as a way of transitioning from a point cloud data set to a topological space, and then using tools of algebraic topology to analyze the data set. The goal of this semester is to develop the mathematical foundations to understand topological data analysis and then to test my learning through some applications. We plan to use ""Computational Algebraic Topology Lecture Notes"" by Vidit Nanda and selections from ""Elementary Applied Topology"" by Robert Ghrist for this semester.
Towards an Introduction to Yang-Mills Theory
Fernando Villegas Negrete
Marielle Ong
The goal of the project is to learn the basics concepts of geometry, topology and algebra that are fundamental for the understanding of mathematical gauge theories, in particular, for Yang-Mills theory. The plan is to begin with an study of Lie groups and Lie algebras, Fibre bundles, Connections and curvature and then to either Spinors and the development of the mathematical physical theory of Yang-Mills , or to a more algebraic-geometrical perspective. Or both!
Weak Assumption Black-Scholesness
Henry Johnson
Hunter Stufflebeam
Over the course of the semester, we hope to gain a deeper understanding of the mathematics behind the Black-Scholes option pricing model and it's counterparts and various proofs. Then we will investigate the accuracy the random walk returns assumption and attempt to characterize the relationship between deviations from that assumption and the structure of the resulting model.
What Knot to Do When Synthesizing Medications: The Link Between Topology and Chirality of Organic Molecules (video, slides)
Anna Nguyen
Yi Wang
"We plan to learn an introduction to topology, specifically knot theory and its applications to determining the stereochemistry of an organic molecule. This study will commence with an introduction to knot theory including ambient and planar isotopy, projections via Reidemeister moves, and chirality. Then, chirality from a chemical lens and relationships between the geometry of organic molecules to knot theory will be explored. We will conclude with an emphasis on the chirality of modern medicine including the Thalidomide scandal and current FDA approved COVID-19 treatment, an RNA Polymerase Inhibitor called Remdesivir.
We plan to use The Knot Book by Colin Adams and When Topology Meets Chemistry by Erica Flapan. Supplemental materials include organic chemistry videos by Khan Academy, medical manuscripts online, and learning from Penn Chemistry Professor Amos Smith's CHEM 242: Organic Chemistry II class this semester."
Spring 2021 Projects
A Crash Course on Homotopy Theory (video)
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 $\mathbb{R}$ (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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)
Adam Zheleznyak
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)
Nick 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.
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)
Adam Zheleznyak
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
Hadi Elzayn
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.