## Directed Reading Program

The Directed Reading Program (DRP) pairs undergraduate students with graduate student mentors to undertake independent study projects of various sizes and scopes over the course of the spring semester. The projects can take the form of reading and working through a mathematics text, reading research papers, or even doing research.

The goal of the DRP is to enable undergraduate students to study mathematics in greater depth than is possible in a classroom, to increase interaction between undergraduates and graduates, to give undergraduates an opportunity to practice explaining mathematical ideas in conversations and in presentations, and to give graduates an opportunity to share their passion for mathematics.

For questions about the program, please email mathdrp@brandeis.edu.

### Structure of the Program

Selected students are expected to meet with their mentors for at least one hour each week to discuss their progress, put in at least four hours of independent work between meetings, and give 20-minute presentations on some aspect of their work at an end-of-semester gathering for all participants.

#### Undergraduate Students

Any sophomore, junior, or senior who has taken 15A (applied linear algebra) and 20A (calculus of several variables) is eligible to apply. First-years who have seen this material are considered on a case-by-case basis. Acceptance into the DRP is determined by previous coursework in mathematics (including final grades) and availability of mentors. Up to five pairings are made. Students receive a small budget for textbooks used, but no course credit is awarded. Students with heavy or challenging course loads should think carefully before committing to the DRP.

#### Graduate Mentors

Any graduate student who has passed the teaching apprenticeship program can apply to be a mentor. Each mentor is expected to guide his or her student through the study of a topic. This means helping the student come up with a study plan. This also means meeting with the student every week to answer questions, point out subtleties, explain the big picture, and have the student present material. Each mentor is also expected to assist with the presentation at the end of the semester by helping with the outline of the talk, having the student give practice talks, and helping with LaTeX if the student wants to give a beamer presentation. Mentors are modestly compensated for their work.

### Spring 2019 Program

There are eight graduate student mentors offering eight projects. Up to five projects will run. You can apply online. The deadline to apply is **Monday, January 21, 2019**. Each participating student will give a 20-minute presentation on their project at the end of the spring semester (date to be announced). Pizza and other refreshments will be served; all are welcome to come.

#### Project descriptions, Spring 2019

Category theory provides a structure and language in which to study all other types of mathematics. One has the category of sets, the category of groups, the category of abelian groups, etc. In studying general category theory one obtains new techniques for proving results that were otherwise out of reach or incredibly difficult. It allows us to study the relationship between seemingly different structures, such as geometric things and algebraic things. Sometimes, category theory inspires a new question whose answer has nothing to do with category theory in general but is of great use. In short, knowing a bit of category theory can help in almost every field of mathematics.

**Suggested Book(s):**

*An Introduction to Category Theory*by Harold Simmons- More advanced texts as determined by student interest

Algebraic Varieties are the central object of study in Algebraic Geometry. A variety is the set of common zeroes of a collection of polynomials. The aim of this project is to understand Hilbert Nullstellensatz for Affine Varieties. We will start with a review of polynomial rings and field theory. After doing some commutative algebra including Hilbert Basis theory and Noether normalization, we will focus on Theory of Affine Varieties and Nullstellensatz. If time permits, we will do lots and lots of examples together and some generic properties of varieties.

**Prerequisites:**

Some knowledge of rings and fields is essential, knowledge of pointset topology would be helpful but not essential.

**Suggested Book(s):**

*Introduction to commutative algebra* by Atiyah and MacDonald.

Cohomology theories arise naturally in many contexts in algebra and are a important tool in all branches of algebra. Homological algebra is the branch of mathematics that studies (co)homology in a general algebraic setting. The formalism of derived functors is an abstraction of all such theories. In this project, we will learn about derived functors and their instances like Ext, Tor and group cohomology. Depending on time and interest, we may also take a brief look at spectral sequences and their applications.

Homological algebra is a useful tool to understand many of the algebraic behaviors of abelian categories. Instead of learning such properties case by case, we will learn the description of those properties in the language of categories. The language of category theory will not only provide a new aspect of those algebraic properties in a general setting, but also give some specific new results. This course will be helpful for students who want to become a math graduate students working in topology, algebra, or number theory. This course will cover additive and abelian categories, derived functors, sheaves, and cohomology (including group cohomology, simplicial cohomology, and sheaf cohomology).

**Prerequisites:**

Algebra 1 (Math 131a) or equivalent (Basic properties of groups and rings).

**Suggested Book(s):**

1. *Sur quelques points d’alg`ebre homologique* by Alexander Grothendieck

2. *Algebraic Topology* by Allen Hatcher

3. *An Introduction to Homological Algebra* by Charles A. Weibel

The mathematical study of knots and links began in the 19th century when physicists proposed that knotted and linked vortices in the aether might provide an accurate model for atoms and molecules. Although the Michelson-Morley experiment in 1887 effectively disproved the existence of the aether and put paid to this physical interpretation of knot theory, the early 20th century and the birth of topology brought to light deep connections between knot theory and the study of three and four dimensional manifolds. Many questions about 3- and 4-manifolds can be reformulated into questions about the existence and properties of certain knots and links and these questions have far reaching implications in physics and other areas of mathematics. The last fifty years have seen an explosion of beautiful theorems about knots and links, several of which have earned Fields Medals for their authors, and knot theory is now a prominent, active area of research. Dale Rolfsen's book "Knots and Links" gives an exposition of the classical theory of knots and provides the background needed to understand the discoveries of the last fifty years. The aim of this reading program will be to learn the major theorems detailed in this book and to understand how knot theory interacts with the other areas of mathematics.

Suggested Book(s):

1. *Knots and Links* by Dan Rolfsen

In this reading program we will explore the magnificent world of partial differential equations. Starting with the classical equations: transport equation, harmonic equation, heat equation, and wave equation, we will explore the basic ideas of how people approach and perceive PDEs of different types. Later we shall focus on the theory of elliptic PDEs, which plays a fundamental role in a lot of modern geometry theories. The goal is to understand weak solutions, regularity, and maximum principle etc. of such PDEs.

Suggested Book(s):

1. *Partial Differential Equations* by L. Evans

2. *Elliptic Partial Differential Equations of Second Order* by D. Gilbarg and N. Trudinger

One of the central problems in algebraic number theory is to study the extensions of number fields, to which one of the methods is to study the ramification properties of the extensions. In chapter V of Kato’s book, one sees that the desired ramification properties of an abelian extension over the rational number field Q can be achieved via looking into a certain type of known fields, the cyclotomic fields, hence class field theory is involved. Roughly speaking, class field theory is the generalization of the above idea to any abelian extension of number fields, instead of focusing on only abelian extension over Q. It can be stated in different ways while the goal of this project is to understand the one using the Artin map. We will first work with some motivating examples in number theory, such as the two square theorem, then turn to the general theory of number fields. After that, we will explain the significance of the examples with the theory and introduce class field theory.

Prerequisites:

Familiarity with rings, field extensions, and the concept of Galois groups is suggested, but can be regarded as part of the project, hence not required.

Suggested book(s):

1. *Number Theory 1: Fermat’s Dream* by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito.

2. *Number Theory 2: Fermat’s Dream* by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito.

3. *Algebraic Theory of Numbers* by Pierre Samuel.

** **

Riemann surfaces have a wide range of connections with modern subjects in mathematics, and is interesting as a subject itself. During the semester we will start with the basic definition with a mimimal background of complex analysis, then move to applications in differential geometry and topology. The final goal is to introduce the uniformisation theorem, Riemann-Roch thoerem, and moduli space of Riemann surfaces.

**Suggested Book(s):**

1. *Riemann Surfaces* by S. Donaldson