## Johns Hopkins Chapter

### What is DRP?

The Directed Reading Program (DRP) is a program that pairs undergraduate students with graduate students for one-on-one independent studies over the course of a semester. The program was started at the University of Chicago but it is now running in several mathematics departments in the country.

The program is largely free form and without theme inasmuch as content is concerned. To this end, both graduates and undergraduates commonly propose study material and directions. Nevertheless, for comparison and inspiration, a list of past projects and descriptions may be found below. We have also arranged with the university for a special, 1-credit course associated with the program in which participants may elect to enrol, and selected mentees will have their chosen textbook (or similar resource) bought for them by the department.

If you are interested, please submit a single application using this form by Monday the 8th of February. For more information, feel free to contact the organisers at drp@math.jhu.edu

### 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.

### Presentations

At the end of the semester there will be a presentation session. All members of the department and friends of speakers are welcome to join. There will be pizza!

# Fall 2019 Projects

Class Field Theory
Xiyuan Wang
Zechen Yi
We will try to understand the statement of class field theory. And we do some basic computations about Hecke characters and Galois characters.
Reproducing Kernel Hilbert Spaces
Patrick Martin
James Sherwood
The goal of this semester is to investigate Reproducing Kernel Hilbert Spaces (RKHS) and their application to Machine Learning (ML). Beginning with the foundation of Hilbert Spaces, I will study the importance of Reproducing Kernels and the role of the Riesz Representation Theorem. Afterwards, I will narrow the focus to RKHS’s utilized within ML, ending with the nuances of their uses.
Graph Theory
Ben Dees
Georgie D’Sanson
We will investigate aspects of graph theory in this project, including standard notions and invariants such as crossing number and chromatic number. From there, there are many options open: we may explore bounds on these invariants, coming from a variety of places, or investigate other areas such as applications of graph theory or the theory of infinite graphs. The main reference for this will be West’s Introduction to Graph Theory, with other sources used to provide context as necessary.
Partial Differential Equations
Letian Chen
Casey Weiner
Abstract: We will be learning some basic theory about partial differential equations (PDEs) following “Partial Differential Equations: An Accessible Route through Theory and Applications” by András Vasy. We will begin with (nonlinear) first order equations and turn to weak solutions to second order linear equations. Towards the end, we will be studying some Fourier transform methods in solving PDEs.
Category Theory
David Myers
Wonjun Yoon
We will be learning the theory of categories with an eye to their use in homological algebra. In particular, we’ll be reading though Harold Simmons’ “Introduction to Category Theory”. Our goal is to have an understanding of the notion of universal property and universal construction, as well as the Yoneda Lemma that undergirds the theory.
Representation Theory
Hanveen Koh
Wialliam Bernardoni
We’re interested in the finite dimensional representations of semisimple Lie groups and Lie algebras over the complex field. We’ll first learn about basic concepts for Lie groups, Lie algebras, and their relationship, and later concentrate on examples such as representations of sl_2(C) or sl_3(C). The main reference is “Representation Theory: A First Course” by Joe Harris and William Fulton.
Bayesian Inference
Daniel Fuentes-Keuthan
Deepan Islam
This semester we will study a bayesian approach to statistical learning. Much of the difficulty of such an approach involves computing distributions given by computationally intractable integrals arising as posterior distributions through Bayes law. We will study two classes of approximations to these integrals: the deterministic approximations, which seek to approximate integrals by finding an exact solution to an approximation, and stochastic approximations such as Markov chain Monte Carlo methods, which approximate the solution using randomness and the law of large numbers.
Real and Complex Analysis
Caroline VanBlargan
Cornell Holmes
We will be working through Rudin’s Real and Complex Analysis, with a focus on studying some topics in Complex Analysis. After going through the more foundational aspects of complex analysis and holomorphic functions, we will read through topics on harmonic functions, the maximum-modulus principle (including the Phragmen-Lindelof Method), theorems on approximating holomorphic and meromorphic functions with rational functions, and conformal mappings. If time allows, we will cover theorems related to the zeros of holomorphic functions, such as the Weierstrass Factorization Theorem.
Scratching the Surface of Number Theory
Keaton Stubis
Site Gao
Our goal will be to discuss some interesting ideas one can encounter early on in number theory. The plan is to start with group theory (and maybe a bit of additional abstract algebra), and then eventually build up to Galois theory of finite extensions of $$\mathbb{Q}$$. Along the way we will spend a session discussing quadratic reciprocity, and may add other small “punctuations”.
Lattice Theory
tslil clingman
James S. Wang
In this project we will aim to understand the fundamentals of lattice and order theory. In particular, once the fundaments have been explored we will direct our attention to the notions of ideals and filters, complete lattices, and distributive lattices. Time allowing, the connections with topology and logic will discussed. The main reference for this project is the book of Priestley and Davey, “Introduction to Lattices and Order”.
Introduction to Differential Geometry
Jeff Marino
Samuel Salander
In this project we’ll explore the geometry of regular curves and surfaces. Our goal is to develop the material needed to understand mathematical relativity. To that end, we’ll pay special attention to the use of coordinates, an intrinsic approach to geometry, and a simultaneous development of tensor calculus.
Cyclotomic polynomials and algebraic number theory
Kalyani Kansal
Tongtong Chen
In this project, we plan to study Galois theory and basic Algebraic number theory. We will attempt to understand the rings of integers for number fields, motivated by the study of cyclotomic polynomials and pell equations. We will also work on related problems to reinforce understanding.

# Spring 2019 Projects

Homotopy Type Theory
tslil clingman
Nandan Kulkarni
Our goal in this project is to understand the fundamentals of Martin-Löf type theory with intensional equality and univalence, so that we might observe the natural emergence of both logic and topology in this framework. In particular we will aim to understand the basic type formers and theorems underlying the framework, as well as the implications of the univalence axiom. From this we aim to develop intuitions about constructive mathematics through the propositions-as-types interpretation, as well as to develop algebraic notions of topology as encapsulated in higher inductive types.
Smooth Manifolds and Riemannian Geometry
Cheng Zhang
Zecheng Yi
We will study differential geometry with “Introduction to Smooth Manifold” and “Riemannian Geometry” as the reference textbooks. We will first go through some basic definitions and theorems concerning smooth manifolds and smooth maps between manifolds. This includes a discussion of submersions, immersions, and embedding. With the basic knowledge in differential geometry, we will go on to consider some topics on Lie groups, which are both groups and smooth manifolds. Our final goal is to cover vector bundles, tensors, and differential forms, which play important roles in study of algebraic topology and algebraic geometry.
Point-Set and Algebraic Topology
Hannah Wool
In this project, we’ll explore beyond point-set topology into the area of algebraic topology, including homotopy of paths, covering maps, Seifert-van Kampen Theorem, homology of surfaces, etc. The main reference is “Munkres&rsqpo; Topology”.
Fourier Analysis
Xiaoqi Huang
Michael Farid
We are going to focus on Fourier analysis, including Fourier series and Fourier transform, and their applications. If time allows, we will also cover some other transforms like discrete Fourier transform or wavelet transform , and some analytic number theory using Fourier transform. The main reference is Stein’s book : Fourier Analysis, an introduction.
Algebraic Geometry
Daniel Fuentes-Keuthan
William Bernardoni
We will be studying an introduction to algebraic geometry using the book “Undergraduate Algebraic Geometry”. The goal is to learn fundamental topics like the nullstellensatz and to hopefully see some applications to classical geometry projects. We will also learn some background commutative algebra.
Category Theory for Applications
David Myers
Benjamin Fried
We will learn the language of categories, functors, and natural transformations as a general framework for the study of complex systems and how they combine. We will begin with the theory of orders and the “generative effects” that are found in order maps, and work our way through monoidal categories as “resource theories” on to the notion of an operad as an abstract setting to frame questions about combining systems. Our main resource will be Fong and Spivak’s “Seven Sketches in Compositionality”.
Partial Differential Equations
Jeff Marino
Joe Klein
The study of PDE at a rigorous level is a milestone in an analyst’s training. In this project we will explore several topics in Evan’s textbook on the subject. First we will compare and contrast Laplace’s equation, the heat equation, and the wave equation; this “trophy case of PDE” highlights the different features and physical interpretations of linear elliptic, parabolic, and hyperbolic equations. Next we will discover the challenges of nonlinearity by considering the method of characteristics and conversation laws. Time-permitting, we will conclude the program by briefly introducing Sobolev spaces and discussing their role in solving more general PDE.
Functional Analysis
Ben Dees
Dan Swartz
Functional analysis uses techniques of both analysis and linear algebra to study various spaces of interest, especially function spaces. We will be following Kreyszig’s “Functional Analysis” to initially learn about Banach and Hilbert spaces. After that, we will either go through some of the standard theorems of functional analysis or delve into some of the mathematics behind quantum mechanics.
Tensor Analysis and Applications
Quanjung Lang
Michael Crockett
We will study “Tensor analysis and applications” with “Tensor Analysis, Spectral Theory and Special Tensors, Liqun Qi, Ziyan Luo” as the reference textbook. We will first learn some basic definitions and theorems about tensors and tensor decomposition, for example, CP decomposition, Tucker decomposition and different eigenvalues. Then we will follow the paper “On the Expressive Power of Deep Learning: A Tensor Analysis” to prove the improvement of Hierarchical Tucker (HT) decomposition to CP decomposition. And we will also show this improvement by an example of pattern recognition.
Understanding $$SL_2(\mathbb{Z})$$
Xiyuan Wang
Sally Bao
In this project, we would like to study the linear fractional action of $$SL_2(\mathbb{Z})$$ on the complex upper half plane. We will first try to understand the special linear group $$SL_2(\mathbb{Z})$$. Then we want to study the orbits and stabilizers of this group action. Finally, we want to understand the fundamental domain associated to this group action.
Topological data analysis
Patrick Martin
Tiffany Hu
We will be investigating topological data analysis, the study of how to leverage topics from topology to discover structure in data. In addition to learning about various tools in the field, such as Voronoi diagrams and persistent homology, we will also pay attention to how these tools are implemented for practical use.

# Fall 2018 Projects

Hamiltonian Mechanics and Symplectic Manifolds
Sydney Timmerman
This directed reading project will develop the basics of differential geometry and Lie groups, with the goal of developing the background of Hamiltonian mechanics and symplectic manifolds. We will begin with a survey of manifolds and smooth maps of manifolds, and then explore vector and tensor fields on manifolds. Time permitting, we will progress into exterior algebra and calculus on forms. Background on the geometry of Lie groups and their action on manifolds will be filled in as necessary to progress into Hamiltonian mechanics and symplectic manifolds. The primary reference is Differential Geometry and Lie Groups for Physicists by Marián Fecko.
Partial Differential Equations
Ben Dees
We will be studying partial differential equations, primarily using the text Partial Differential Equations by Lawrence Evans. We will begin with some classical partial differential equations, and then progress to a more general setting. This may include topics such as Sobolev spaces, second-order elliptic equations, or Hamilton-Jacobi equations, and may touch on various methods of finding solutions including Fourier analysis.
Elliptic Curves
Daniel Fuentes-Keuthan
Brian Wen
We will be studying the connections between algebra, number theory, and geometry by learning about elliptic curves and the various algebraic prerequisites needed to understand them. We will work through Tate and Silverman’s Rational Points on Elliptic Curves with a goal of understanding Mordell’s theorem.
Generating Functions
Jeffrey Marino
Raymond Weisbrot
Generating functions allow us to solve combinatorial problems using methods of analysis and algebra; their theory is a celebrated link between discrete and continuous mathematics. In this project we will discover the role that these objects play in combinatorial discourse. First we will address the fundamentals: the basic principles of counting, as well as the standard operations one uses when employing generating functions. Next we will dive into applications, the selection of which is to be determined. Some possible topics include the Catalan numbers, the Stirling numbers, and the exponential formula. Our primary reference is Boná’s A Walk Through Combinatorics, with supplementary reading from Wilf’s renowned Generatingfunctionology.
Introduction to Nonstandard Analysis
Michael Patrick Martin
Jeffrey Zhang
We will be studying nonstandard analysis from the text Lectures on the Hyperreals, by Robert Goldblatt. We will begin by understanding the hyperreals and their correspondence with the reals through the transfer principle. We will then move towards reconstructing calculus and basic analysis through this lens. Finally, we hope to see some applications of nonstandard analysis such as Loeb measure and in Ramsey Theory.
Category Theory
tslil clingman
Chase Fleming
The focus of this the project will be the establishment of the underpinnings of the general theory of categories. We will begin the project by exploring the notions of category, functor, natural transformation, limit, and adjunction -- each in the presence of interesting and varied examples. Throughout our work we will take special care to interpret each new concept simultaneously as a specific instance of previous established notions, and as a generalisation thereof, thereby witnessing the mantra that “all notions are examples of all others”. Another theme that we will aim to introduce is that of categorification -- arguments made should, in structure-related cases, be rendered independent of the particulars of the objects at hand, and thus readily generalise to various new settings. Time allowing and interest dictating, we will investigate the formalism of monoidal categories, monads, and more exotic categorical structures beyond (2-categories, enriched categories, double categories, internal categories) with the expectation that our established theory is a specialisation of a yet broader approach.
Algebra: Chapter 0
Xiyuan Wang
Yi Hong
We will be studying category theory and abstract algebra. The main topic are groups, rings, fields and modules. We will also try to understand these algebraic structures from the point of view of category theory. The main reference is Algebra: Chapter 0 by Paolo Aluffi.

# Spring 2018 Projects

Topology and Data: An Introduction to Persistent Homology
Thomas Brazelton
Mira Wattal
This project will provide an overview of persistent homology, one of the major theories in the growing field of topological data analysis. We will begin with an introduction to the study of modules, CW complexes, and cellular homology. We will then discuss the Rips complex and Cech complex, and explore persistent homology and barcodes. Given time, we will discuss Morse filtrations, metrics on the space of persistence diagrams, and discuss direct applications of persistent homology such as 3D image reconstruction.
Arithmetic function on $$\mathbb{Z}[\sqrt{2}]$$
Xiyuan Wang
Raymond Weisbrot
We will be learning analytic number theory from the text Analytic Number Theory for Undergraduates by Heng Huat Chan. One important topic of analytic number theory is the arithmetic function on $$\mathbb{Z}$$ and its L-function. As a final project, we want to study the arithmetic function on $$\mathbb{Z}[\sqrt{2}]$$.
Introduction to Manifolds
Eric Cochran
We will be studying the theory of manifolds starting from basic multivariable calculus with a goal towards understanding Lie Groups and Lie Algebras. The primary reference is Loring Tu’s “An Introduction to Manifolds”.
Differential topology and related topics
Shengwen Wang
Chris Chia
Differential topology is dealing with smooth functions on manifolds and differentiable maps between smooth manifolds. We will start from the book "Topology from a Differential Viewpoint" by Milnor. We will first go over the non-algebraic-topology proof of Brouwer fixed point theorem using Sard’s theorem, degree theory of smooth maps, and further applications depending on time. Our overall goal is to gain fluency in the language of differential topology.
Braid group representations and Knot invariants
tslil clingman
Robert Barr
We will be studying knot theory with an emphasis on modern knot invariants, algebraic quantities of interest that are the same for equivalent knots, which began with the discovery of the Jones polynomial in the ’80s. First we will review elementary knot theory and the theorems of Alexander, Markov, and Artin which together allow knots to be studied from the perspective of braid groups. With this foundational knowledge established we will move to explore two general families of braid representations and their corresponding invariants obtained from Hecke and Temperley-Lieb algebras, including specifically the Burau representation and HOMFLY polynomial. With this established we will be in a position to study representations obtained from the Yang-Baxter equation. Finally, as time and interest permit, we will investigate the theory of quantum groups and ribbon categories, an important source of R-matrices.
Goodwillie Calculus
Daniel Fuentes-Keuthan
Aurel Malapani-Scala
The purpose of this directed reading will be to develop the rudiments of the calculus of functors, also known as Goodwillie calculus. We will begin with a brief review of some prerequisite material, namely topological spectra, homotopy colimits, the Freudenthal suspension theorem, and the Blakers-Massey Theorem. The main references for these topics will be Stable Homotopy and Generalized Homology by J.F. Adams, A Primer on Homotopy Colimits by Daniel Duggar, and Cubical Homotopy Theory by Brian A. Munson and Ismar Volic. Once acquainted with these topics, we will move on to reading and understanding Thomas G. Goodwillie’s paper, Calculus III: Taylor Series.
Noncommutative Algebra
Hanveen Koh
Jin Lu
We will be learning ring and module theory with an emphasis on noncommutative algebra. We will first look at the interplay between the structure of a ring and the structure of modules over that ring. Our main interest is the classification of finite dimensional central division algebras over a given field, and it will lead us to the Brauer group which has close ties with algebraic geometry and number theory. Our primary reference is Noncommutative Algebra by Benson Farb and R. Keith Dennis.
Introduction to Mathematical Control Theory
Patrick Martin
Julia Costacurta
We will be learning about optimal control theory, primarily from the text Introduction to Optimal Control Theory by Macki and Strauss. We will begin with background and motivation, and then move to a thorough treatment of the linear autonomous case. Then, we will extend what we have learned to cover characteristics of general optimal control problems.

# Fall 2017 Projects

Fundamentals of general topology
tslil clingman
Alex Cornell Holmes
We will be pursuing an understanding of the fundamentals of general topology from an axiomatic standpoint, beginning with the elementary definitions and working our way towards the separation axioms and compactness. Throughout this process, emphasis will be placed the utility of categorical and lattice theoretic concepts as a clarifying, unifying and generalising framework. Time and interest allowing, we will attempt to broaden the scope of our discussion by looking at such topics as uniform spaces and frames. Regardless of stopping point, the overarching goal is a well-grounded fluency in the language of general topology. Our reference material is "General Topology" by S. Willard and "Counterexamples in topology" by L. Steen and J. A. Seebach, Jr.
Real algebraic geometry
Daniel Fuentes-Keuthan
Elvin Xiaoqiang Meng
We will be learning real algebraic geometry from the text Real Algebraic Geometry by Bochnak-Coste-Roy. Real algebraic geometry is the study of subsets of a real ordered field defined by the zero sets of polynomials, as well as those regions where polynomials have constant sign. Working over non-closed, ordered fields leads to certain intricacies which have turned out to be useful in solving certain long standing problems. Our goal for this semester will be to explore classical real algebraic geometry, leading to a proof of Hilbert’s 17th problem.
Knot theory and its applications