Directed Reading Program
Program Overview
The Directed Reading Program pairs undergraduate students with graduate student mentors to read and work through a mathematics text over the course of one semester. The pairs meet once each week for one hour, with the undergraduates expected to do about 4 hours of independent reading per week. At the end of the semester, undergraduates either give a talk to their peers or prepare a short exposition of some of the material from the semester. Undergraduates are expected to have a high level of mathematical maturity and eagerness to learn the topic.
Note that for graduate students with more than one project listed, not all projects may be offered.
Who is Eligible?
The program is aimed at undergraduate mathematics related majors, and could also be suitable for undergraduates with mathematical interests who would like to further explore the field. We particularly encourage applications from women and members of underrepresented minority groups.
Why You Should Participate?
 You like math and have considered a mathematics major, but want to learn more about what math “is”.
 You know that the grad students are hiding in their cubicles all day and are doing mysterious cool stuff. This is your opportunities to know more about them mathematically and become friends with them!
 Research in some field in mathematics seem really cool, but you really don’t have the foundational knowledge for some fields.
 Studying by yourself seems really daunting and now you can start with a group to effectively learn the material under older math friends’ guidance and get plenty of chance to explain the material to someone else!
How to Apply
The deadline for applying to be a mentee is 11:59pm on Friday August 31, 2018. Please read the project description and fill out the application form here.
Questions
If you have questions about the program or the specific projects that you are interested in, feel free to talk to Shiyue (shiyue.li@yale.edu, the organizer of DRP Fall 2018) or any related graduate student mentors in the project descriptions!
2016
Spring

Graduate Mentor:Book:Anthony W. Knapp, Lie groups beyond an introductionDescription:
We start with the basic definitions of Lie groups and Lie algebras. We then follow with basic representation theory of Lie groups and Lie algebras, and structure theory of Lie algebras and root systems. The goal is to build a good knowledge of general Lie groups

Graduate Mentor:Book:Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculusDescription:
The aim of this project is to become familiar with two of the main concepts in probability theory, namely Markov processes and martingales. Our main example of both concepts will be Brownian motion in Rd. One of the main applications of the notion of martingales is its connection to partial differential equations, which leads to the study of integration with respect to stochastic processes and in turn to the study of socalled stochastic differential equations.

Book:Enrico Bombieri and Walter Gubler, Heights in Diophantine geometryDescription:
This project is a rigorous introduction to modern arithmetic geometry. No previous exposure to algebraic geometry is required, although that would be helpful. We will start with the appendix on algebraic geometry and set up the Weil heights in the opening two chapters and then, depending on the student’s interests and background

Book:John Baez and Javier P. Muniain, Gauge fields, knots and gravityDescription:
The first few chapters of the book cover basic differential geometry, including the theory of manifolds, vector fields, and differential forms. These concepts are used to formulate Maxwell’s equations on arbitrary spacetime manifolds. The second part of the book presents the theory of vector bundles and connections and uses these concepts to discuss gauge theory and its relation to knots. The final part of the book explains Riemannian geometry and its applications in general relativity. Ideally, I would like to at least get to the section on knot theory, but in principle we could stop anywhere and it would still be a satisfying experience for the student. Actually, I think there’s a danger that we might finish the book too soon. If that happens, there are plenty of online materials and texts that I can share with the student.

Book:John Baez and Javier P. Muniain, Gauge fields, knots and gravityDescription:
Statistical Mechanics is a big branch of modern physics that studies properties of macroscopic systems, typically having large number of degrees of freedom, for example, gases or fluids. In such situation as one can suspect it is too difficult to give precise answers about microscopic behaviour of systems. How ever, some macroscopic characteristics still can be investigated and that amounts to finding certain probability distributions.

Book:Rodney J. Baxter, Exactly solved models in statistical mechanicsDescription:
Statistical Mechanics is a big branch of modern physics that studies properties of macroscopic systems, typically having large number of degrees of freedom, for example, gases or fluids. In such situation as one can suspect it is too difficult to give precise answers about microscopic behaviour of systems. How ever, some macroscopic characteristics still can be investigated and that amounts to finding certain probability distributions.

Graduate Mentor:Book:John Stillwell, Naive Lie theoryDescription:
Stillwell does an amazing job of introducing Lie theory using nothing more. He introduces the abstract group theory and the differential geometry that are needed for the book. Everything he does can easily be understood by following elementary computations.

Book:Brian Hall, Lie groups, Lie algebras, and representationsDescription:The theory of Lie groups and Lie algebras is not only beautiful – it is of great importance in physics, where the representation theory of these objects is used to study the symmetries of the laws of motion and similar.

Graduate Mentor:Book:Terence Tao, Expansion in finite simple groups of Lie typeDescription:
We will follow Tao’s book, based on a course given at UCLA. We will learn about the theory of expander graphs, isoperimetry in graphs, quasirandomness, probability on groups, arithmetic combinatorics, and applications to number theory.

Graduate Mentor:Book:[TV10] Terence Tao and Van H. Vu, Additive combinatorics, [Tao14] Hilbert’s fifth problem and related topics, [Tao08] Product set estimates for noncommutative groups.Description:Arithmetic Combinatorics (also formerly known as Additive Combinatorics) is a very popular area of research. In my opinion, what makes it extremely interesting is the very strong interaction between numerous fields such as: number theory, combinatorics, group theory, harmonic analysis, ergodic theory, and many more. There was tremendous progress in recent years, and both classical and modern viewpoints will be studied. Historically, additive combinatorics studies additive properties of subsets of the integers, or more generally, abelian groups. In the past 10 years, it has been partially extended to the noncommutative case, with many important applications in group theory and number theory. We will start with a quick introduction to additive combinatorics using [TV10] and to move on directly to the noncommutative case using [Tao14] and to some extent, the seminal paper [Tao08]