

What is DRP?
The Directed Reading Program (DRP) is a program that pairs undergraduate students with graduate students for oneonone 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.
If you are interested in participating in the future either as a graduate student mentor or as an undergraduate student mentee, you can talk to Richard Brown, Mona Merling, Apurva Nakade, Daniel FuentesKeuthan or 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 sessionthis 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!
Spring 2018 pairings and projects
 Topology and Data: An Introduction to Persistent Homology
Mentor: Thomas Brazelton
Mentee: 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 $\mathbb{Z}[\sqrt{2}]$
Mentor: Xiyuan Wang
Mentee: 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 Lfunction. As a final project, we want to study the arithmetic function on $\mathbb{Z}[\sqrt{2}]$.
 Introduction to Manifolds
Mentor: Apurv Nakade
Mentee: 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
Mentor: Shengwen Wang
Mentee: 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 nonalgebraictopology 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
Mentor: Tslil Clingman
Mentee: 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 TemperleyLieb algebras, including specifically the Burau
representation and HOMFLY polynomial. With this established we will be
in a position to study representations obtained from the YangBaxter
equation. Finally, as time and interest permit, we will investigate the
theory of quantum groups and ribbon categories, an important source of
Rmatrices.
 Goodwillie Calculus
Mentor: Daniel FuentesKeuthan
Mentee: Aurel MalapaniScala
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 BlakersMassey 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
Mentor: Hanveen Koh
Mentee: 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
Mentor: Patrick Martin
Mentee: 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 pairings and projects
 Fundamentals of general topology
Mentor: Tslil Clingman
Mentee: 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 wellgrounded 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
Mentor: Daniel FuentesKeuthan
Mentee: Elvin Xiaoqiang Meng
We will be learning real algebraic geometry from the text Real Algebraic Geometry by BochnakCosteRoy. 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 nonclosed, 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
Mentor: Apurv Nakade Mentee: Chris Chia
Description: We will be learning the basics of Knot theory (primary reference: The Knot Book by Adams Collins). The first goal is to understand the algebraic invariants associated to knots, like the knot polynomials, and study their connections to topology and other branches of mathematics. The second goal is to understand how these invariants are used to tackle problems in molecular biology and chemistry.
 Spectral Graph Theory
Mentor: Emmett Wynman Mentee: Hamima Halim
We will be learning spectral graph theory from Dan Spielman's lecture notes. We will be driving towards the definitions and core results of expander graphs, Ramanujan graphs, and interlacing polynomials. The 'hard' goal is to work through the linked lecture notes up through section twentyfour, "Interlacing polynomials and Ramanujan graphs," skipping most tangential topics. Our 'soft' goal is to become literate in the subject enough to understand the techniques outlined in the survey "Ramanujan graphs and the solution to the KadisonSinger problem" arXiv:1408.4421.


