Directed Reading Program

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.

There are 8 graduate student mentors and 15 projects offered (please note, however, that if there is more than one project per graduate student mentor, not all projects will be offered).

The deadline to apply is Friday, January 15, 4pm EST, and the application can be found here.

For a more detailed description of the projects or any questions please contact The DRP Spring 2016 organizers pdf

Thomas Hille (thomas.hille@yale.edu)

Lam Pham (lam.pham@yale.edu)

2016

Spring

  • Graduate Mentor:
    Book:
    Terence Tao and Van H. Vu, Additive combinatorics
    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 non-commutative 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, Erdos-Szemer ́edi conjecture, Green-Tao 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 groups
    Description:

    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 Peter-Weyl 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 4-6). 

  • 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 Analysis
    Description:

     The primary goal is to become familiar with the non-commutative 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 processes
    Description:

    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 introduction
    Description:

    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 calculus
    Description:

    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 so-called stochastic differential equations.

  • Book:
    Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry
    Description:

    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