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: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]

Graduate Mentor:Book:Terence Tao and Van H. Vu, Additive combinatoricsDescription:
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 focus on purely additive combinatorics, studying the basic notions of the field, and some of the major results (Szemeredi’s Theorem, ErdosSzemer ́edi conjecture, GreenTao Theorem, Freiman’s Theorem, etc) and techniques involved in proving these (graph theoretic techniques, character theory and Fourier analysis, ergodic theory).

Graduate Mentor:Book:David Applebaum, Probability on compact Lie groupsDescription:
The goal is to become familiar with the anal ysis tools available for treating probability on groups. We will follow Applebaum’s book. Quickly reviewing some basics of Lie groups (Chapter 1), unitary representations and PeterWeyl theory of compact Lie groups (Chapters 2 and 3), before moving on to the main part of the reading which is the study of probability measures on groups (Chapters 46).

Graduate Mentor:Book:Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T)Description:D. Kazhdan introduced Prop erty (T) in 1967. This property has been very useful and has been used among other things by G. Margulis to give the first explicit construction of expander graphs, and many famous lecture notes are available for supplementary reading.

Graduate Mentor:Book:Gerald .B. Folland, A Course in Abstract Harmonic AnalysisDescription:
The primary goal is to become familiar with the noncommutative Fourier transform which is a very powerful tool. There are beautiful theories developed and we will see some of the special cases, in particular in the abelian case and the compact case. We will start with some review of topological groups and functional analysis (Banach algebras, spectral theory). Depending on the familiarity of the student, we can move quite quickly to unitary representations and functions of positive type before getting to our two main cases of study: analysis on locally compact abelian and compact groups.

Book:Richard Durrett, Essentials of stochastic processesDescription:
Stochastic processes constitute an important subject in probability theory and have strong connection with ergodic theory, analysis, theoretical computer science, etc.

Book:Serge Lang, SL2(R)Description:
Stochastic processes constitute an important subject in probability theory and have strong connection with ergodic theory, analysis, theoretical computer science, etc.

Graduate Mentor:Book:Serge Lang, SL2(R)Description:
We will follow Lang’s SL2(R) and it is mainly an introduction through SL2(R) to the infinite dimensional representation theory of semisimple Lie groups. We don’t need any knowledge of Lie theory here.

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