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 2020 Program
There are four graduate student mentors offering four projects. Each participating student will give a 20-minute presentation on their project. Time and location will be announced. Pizza and other refreshments will be served; all are welcome to come.
Project descriptions, Spring 2020
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Graduate mentor: Job Rock
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
Graduate mentor: Tarakaram Gollamudi
Given a quadratic polynomial ax^2+x+c, the roots are given the formula (-b±√(b^2-4ac))/2a. Gerolamo Cardano came up with a formula to solve all the cubic polynomials, and a similar formula can be given for quartic polynomials by Ferrari’s method. The next natural step here is to find formulas to solve all the roots of polynomial of degree 5 or more, or more formally can we write down roots of a polynomial in terms of its coefficients, the four basic arithmetical operations +,-,×,÷ and n th roots (i.e., square roots, cube roots etc), formally we call this as Solvability by radicals. After about 350 years, a French mathematician Ãvariste Galois answered this questions by introducing Galois theory. Galois Theory assigns, to each polynomial, a mathematical structure called a group. In this project we study a necessary and sufficient condition for a polynomial to be solvable by radicals.
Prerequisites:
Basic introduction to theory of Fields and groups would be beneficial.
Graduate mentor: Langte Ma
After choosing a basis for a finite dimensional real vector space, any linear transformation can be represented by a matrix. If the vector space has extra structures, like metric, symplectic structures, the linear transformations preserving those structures correspond to matrices of certain special form. In this reading program, we will follow the book Matrix Groups to study those matrices via explicit examples from the perspective of algebra and topology.
Prerequisites:
The minimal prerequisites are calculus and linear algebra.
Suggested Book(s):
- Matrix Groups by Morton Curtis (Universitext, Spring 1979).
Graduate mentor: Eric Hanson
When releasing sensitive data, it is critical to protect the privacy of those represented in the dataset. It has been shown that simply eliminating obvious identifiers (for example names, social security numbers, and addresses) is not enough to guarantee that someone’s personal data cannot be reconstructed from the released data. Differential privacy provides a mathematically rigorous way to guarantee that a release of information about sensitive data will not violate any one person’s privacy. The key idea is to add noise randomly sampled from a probability distribution to the results of statistical queries. The difficulty comes in making the noise large enough to protect privacy, but small enough to preserve the utility of the statistics. In this program, we will learn about the theory of this framework (and its derivatives) and how it can be implemented for certain statistics. Many of the specific topics will be chosen based on student interest.
Prerequisites:
Math 36a (Probability): differential privacy is based on sampling from continuous probability distributions and often involves conditional probability and the law of total probability.