## 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 programme 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 programme in which participants may elect to enrol, and selected mentees will have their chosen textbook (or similar resource) bought for them by the department.

For more information, feel free to contact the organisers at drp@math.jhu.edu or individually, Richard Brown, Apurva Nakade, Daniel Fuentes-Keuthan, and tslil clingman.

### 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 2018 Projects

Hamiltonian Mechanics and Symplectic Manifolds
Sydney Timmerman
Tensors In 10 Minutes Or Less
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
An Introduction to Nonstandard Analysis
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
CLASSIFICATION OF GROUPS WITH ORDER $$\le 20$$
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