Directed Reading Program

What is DRP?

What is expected of mentees and mentors

Presentations

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 L-function. 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 non-algebraic-topology 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 Temperley-Lieb algebras, including specifically the Burau representation and HOMFLY polynomial. With this established we will be in a position to study representations obtained from the Yang-Baxter equation. Finally, as time and interest permit, we will investigate the theory of quantum groups and ribbon categories, an important source of R-matrices.



• Goodwillie Calculus
  
  Mentor: Daniel Fuentes-Keuthan
  Mentee: Aurel Malapani-Scala
  
  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 Blakers-Massey 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 well-grounded 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 Fuentes-Keuthan
  Mentee: Elvin Xiaoqiang Meng
  
  We will be learning real algebraic geometry from the text Real Algebraic Geometry by Bochnak-Coste-Roy. 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 non-closed, 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 twenty-four, "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 Kadison-Singer problem" arXiv:1408.4421.