### Applications for Spring 2018

Applications from undergraduate students to participate are submitted online via the Undergraduate Application Form.

Students will give 20-minute presentations on their project on Wednesday, April 25 at 5pm in Goldsmith 317. All are welcome to attend.

### Other Directed Reading Programs

Our Directed Reading Program is based on programs at several other math departments. Here is an article on an AMS blog about DRPs, and a list of some of these programs.

- Indiana University
- MIT
- University of California, Berkeley
- University of Chicago
- University of Maryland
- Yale University

### Contacts

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

The program is run by the Graduate Student Representatives with assistance from the Graduate Advising Head Joël Bellaïche, the Undergraduate Advising Head Ruth Charney, the Department Chair Dmitry Kleinbock, and the Undergraduate Representatives Kaiyue He, Elana Israel, and Christopher Simonetti.

## Directed Reading Program

### General Information

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.

### Spring 2018 Program

There are six graduate student mentors offering seven projects. Up to five projects will run. The deadline to apply is **Monday, January 15, 2018**, and the application can be found here. Each participating student will give a 20-minute presentation on their project on **Wednesday, April 25th at 5pm in Goldsmith 317**. Pizza and other refreshments will be served; all are welcome to come.

#### Project descriptions, Spring 2018

#### Number Theory

- Graduate mentor: Tarakaram Gollamudi

- The idea of this project is to learn some introductory number theory and at the same time have sneak view of 𝔭-adic analysis. The first part of the project covers some interesting concepts in basic Number Theory like, primes in certain arithmetical progressions, congruences and residues, Fermat's Theorem and approximation of irrationals by rationals.Second part of this project focuses on 𝔭-adic analysis. 𝔭-adic
numbers are introduced by Kurt Hensel are now well established in the mathematical world with applications beyond mathematics. We can view the 𝔭-adic numbers as directly analogous to the real numbers, where we start with the usual the absolute value and look for a completion of ℚ as a metric space and we get ℝ but there is no reason to view the usual absolute value on ℚ as a given. We start this part of the project with a different absolute value which is 𝔭-adic value and then learn what is it's completion, the next objective is to learn 𝔭-adic mean value theorem and then we will stop with some elementary analysis in ℚ𝔭 There aren't any strict prerequisites for the first part, although knowledge of groups (28A) will be helpful. For the second part it would be very helpful to have some knowledge of metric spaces (for example Chapter 2 and 3 of Walter Rudin's Principles of Mathematical Analysis) and basic Field theory usually covered in 28B.

Suggested books:

1. An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright

2. A Course in 𝔭-adic Analysis by Robert, Alain M.

3. 𝔭-adic Numbers An Introduction by Gouvea, Fernando

#### Algebraic Curves

- Graduate mentor: Abhishek Gupta
- Description: Algebraic geometry is one of the central fields in modern mathematics. This project offers a first introduction to this field via the theory of algebraic curves. The book is a classic and widely considered a great first introduction to algebraic geometry. Even though rather elementary in spirit, this book prepares the reader well for modern algebraic geometry. The prerequisites are minimal: basic properties of rings, ideals, and polynomials, such as is often covered in a one-semester course in modern algebra. We will learn the required commutative algebra as needed. We'll use
*Algebraic Curves*by William Fulton (also available for free online on the author's page).

#### The Theory of Modules

- Graduate mentor: Eric Hanson
- Description: Modules are a generalization of vector spaces, where the scalars come from an arbitrary ring rather than a field. The study of modules often provides information about the ring, and is used extensively in virtually every area of mathematics. Many results from linear algebra generalize to the theory of modules, but several theorems (for example the rank-nullity theorem) no longer hold. We will focus on the theory of modules over both non-commutative and commutative rings. Topics will include free modules, projectives and injectives, resolutions, hereditary rings, and other topics determined by student interest. This subject has a lot of freedom and the curriculum can be tailored based on student background. We'll use
*Modules and Rings*by John Daun. - Prerequisites: Linear algebra and at least one more semester of abstract algebra (groups, rings, and intro to algebra 1 are all sufficient).

#### Mapping Degree Theory

- Graduate mentor: Langte Ma
- Description: The notion of degree is recognized as a beautiful achievement in geometry having many interactions with other interesting mathematical notions, for instance winding number, index, order of singular points etc.. In this reading program we begin with a sketch of history of the development of degree. With such a background we can look at degree in different perspectives to approach some interesting classical results in topology and dynamical system, like Borsuk-Ulam Theorem, Jordan Separation Theorem, Hopf Theorem etc.. We will use an introductory book by Outerelo and Ruiz [1]. A knowledge of multi-variable calculus should be enough for this program.
References[1] Enrique Outerelo, Jesús M. Ruiz, Mapping Degree Theory, Graduate Studies in MathematicsVolume 108, American Mathematical Society, 2009.

#### Matrix Groups

- Graduate mentor: Langte Ma
- Description: Lie groups have been playing a fundamental role in mathematics ever since its discovery. But the abstract formal definition usually obstructs the first-time learners from the appreciation of its beauty. To make a compensation, in this reading program we will abandon the language of differential geometry but begin with a relatively low point of view — calculus and linear algebra via the study of matrix groups which form an important class of Lie groups. Through 2- and 3-dimensional examples of matrix groups we will grasp the core properties for general Lie groups. If time permits, some representation theory can also be discussed. The textbook we are going to use is [1].
References[1] Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups, Oxford UniversityPress, 2006.

#### Introduction to Category Theory

- Graduate mentor: Job Rock
- Description: 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. We’ll use use “An Introduction to Category Theory” by Harold Simmons which will be supplemented when necessary from other places in print and online.

#### Introduction to Category Theory

- Graduate mentor: Ying Zhou
- Description: Category theory is widely used in all areas in mathematics. Through this reading course you will be introduced to what categories are and their basic properties. This course requires no prerequisites beyond group theory though understanding of rings and point-set topology certainly helps because they provide a wide range of examples. This course is especially helpful for those who want to become math grad students. Book: Saunders Mac Lane, Categories for the Working Mathematician. Online Resource: Wikipedia and ncatlab.

#### 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.