

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.
This semester we are running a pilot version of the DRP program at Johns Hopkins, which we hope will continue every semester from now on. 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 or 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 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!
Fall 2017 pairings and projects
Project titles and descriptions will be posted here shortly. Here are the pairings of mentors with mentees:
 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.


