### Applications for Spring 2017

Applications from undergraduate students to participate are submitted online via the Undergraduate Application Form. The deadline is January 20, 2017 in the first week of classes. Students will give a 20 minute presentations on Friday, May 5, 2017.

Applications from graduate students to be mentors in this program are submitted online via the Graduate Application Form. The deadline for mentors was Friday, November 18, 2016.

### 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 Daniel Ruberman, and the Undergraduate Representatives Brandon Shapiro and Kaiyue He.

## 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 undergraduates 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 get-together for all participants.

### Spring 2017 program

There are six graduate student mentors offering seven projects. Up to six projects will run. The deadline to apply was **Friday, January 20, 2017**, and the application can be found here. Each participating student will give a 20-30 minute presentation on their project on **Friday, May 5, 2017**, starting at 5 pm in room 317 Goldsmith. Pizza and other refreshments will be served; all are welcome to come.

#### Schedule of talks, Friday May 5, 2017.

- 5 PM PJ Apruzzese. Topic: TBA
- 5:30 PM Eli Goldner. Topic: Category Theory
- 6 PM Isabella Wu. Topic: Geometric Algebra
- 6:30 PM Jianzhi Li. Topic: Real Analysis
- 7 PM Thomas O'Hare. Topic: Lie Algebras
- 7:30 PM Zhiping Lu. Topic: Real Analysis

### Project descriptions, Spring 2017

#### Elliptic Curves and Cryptography

- Graduate mentor: Tarakaram Gollamudi
- Description: The idea is to cover Chapters 2-6 from the book
*Elliptic Curves: Number Theory and Cryptography*, 2nd edition by L. Washington. Time permitting and if the mentee is mathematically mature, a few more advanced topics can be covered (like Chapters 7, 11, 13). - Prerequisites: Math 100A, Math 100B, preferably Math 131A and 108B and some coding experience.

#### 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 providing 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 generally but is of great use. In short, knowing a bit of category theory can help in almost every field of mathematics. We'll use "
*Conceptual Mathematics: A First Introduction to Categories*" (2nd Edition!) which will be supplemented when necessary from other places in print and online. The book starts with real-world scenarios before moving into abstract realm of categories.

#### Measure Theory and Integration

- Graduate mentor: Langte Ma
- Description: The project will be based on the book
*Analysis*by E. Lieb and M. Loss. We shall start with measure theory and integration theory, and move to integral inequalities. Ideally we can also work on further topics like the Fourier transform, theory of distributions, and Sobolev space depending on time and the math background of the mentee.

#### Khovanov Homology

- Graduate mentor: Biji Wong
- Description: The goal of this project is to learn the basics of Khovanov homology, an important and fairly new algebraic tool used to study knots in 3-space. We will follow
*Paul Turner's lecture notes on Khovanov homology*, but we will start with background material on knots and the Jones polynomial, a precursor of Khovanov homology. - Prerequisites: Point-set topology and linear algebra.

#### Lie Algebras

- Graduate mentor: Biji Wong
- Description: This project is an introduction to Lie algebras, which are vector spaces with an internal multiplication. The goal is to understand the classification of semisimple Lie algebras. Our main reference will be
*Karin Erdmann's friendly Lie algebras book*aimed at undergraduates. - Prerequisites: Groups and rings

#### Real and Complex Analysis

- Graduate mentor: Shahriar Mirzadeh
- Description: Real and complex analysis are branches of mathematics that have a lot of applications in both pure and applied mathematics. Measure theory, integration, functional analysis, etc. are topics that are used in practical fields like probability theory, finance, economy, physics, computer science, etc. So regardless of future goals of the mentee and their mathematics backgrounds they will find these topics very useful. There are many excellent textbooks written in the subject that can be used. The classical book "Real and Complex Analysis" written by Walter Rudin is one of the best textbooks that covers topics in both real and complex analysis. However, we can work with each student to choose the best textbook for them based on their background and interest. We will start with Lebesgue integration and measure theory. Then we will learn about Hilbert spaces and other basic topics in Functional analysis. If time permits we will also study Fourier transforms, basic properties of holomorphic and harmonic functions, conformal and other related topics in complex analysis.
- Prerequisites: Undergraduate analysis. Knowing the materials covered in the first seven chapters of
*Principles of Mathematical Analysis*by Walter Rudin is recommended.

#### Geometric Algebra

- Graduate mentor: McKee Krumpak
- Description: This project would focus on the use of algebraic tools (group theory, linear algebra, etc.) in geometry. Topics covered might include: affine geometry, projective geometry, the algebra of quadratic forms, symplectic and orthogonal geometry, Clifford algebras. The primary reference for the project will be
*Geometric Algebra*, by Emil Artin. - Prerequisites: Linear algebra, basic group theory

### Undergraduate Students

Any sophomore, junior, or senior who has taken 15A (applied linear algebra) and 20A (calculus of several variables) is eligible to apply. Freshmen 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 six 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.