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 (Multi-Variable Calculus) 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. 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 2022 Program
There are four graduate student mentors offering six 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 2022
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Graduate Mentor: Shujian Chen
A reflection is a function in the real vector space that sends a vector to its mirror image with respect to a n-1 dimensional subspace. For example, reflections in R^2 are determined by lines through origin. A reflection group is a group generated by reflections. In this program, we will read about various reflection groups and their properties.
Book suggestion: Reflection Groups and Coxeter Groups by Humphreys
Suggested Book(s):
- Reflection Groups and Coxeter Groups by Humphreys
Graduate mentor: Katie Ragosta
Geometric group theory explores the connections between algebraic and topological structures by studying group actions on spaces. In this course, we'll give a brief introduction to geometric group theory. We’ll cover relevant background such as group presentations and the Cayley graph as needed, and we’ll discuss some classes of groups that appear frequently in geometric group theory such as Coxeter groups, Artin groups, and braid groups, as well as spaces they act on. Additional topics can be chosen based on student interest.
Suggested Book(s):
- Office Hours with a Geometric Group Theorist
Graduate mentor: Kewen Wang
Homological algebra is a useful tool to understand many algebraic behaviour of abelian category. Instead of learning those properties cases by cases, we will learn the description of those properties in the language of category. 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 outcomes. This course will be helpful for student who want to become a math graduate student working on topology, algebra, or number theory. The prerequisite for this reading course include the ”Algebra 1” course (knowing the basic property of groups and rings). This course will cover the material including the additive and abelian category, derived functor, sheaf, and cohomology (including group cohomology, simplicial cohomology and sheaf cohomology).
Suggested Book(s):
- Algebraic Topology by Hatcher, An Introduction to Homological Algebra by Weibel
- "Sur quelques points d'algèbre homologique" by Grothendieck
Graduate mentor: Jiajie Zheng
In this reading program, we will read about probability measure theory and ergodic theory. First we will briefly introduce finite measures and why we need them for a rigorous theory of probabilities. As a first application, we will talk Markov chains and their existence (in a probability theory sense). Stationary Markov chains are ergodic dynamical systems that exhibit a very nice mixing property–independence! Then we will talk about other measure-preserving dynamical systems and their mixing properties. We will discuss its applications in metrical number theory, including Diophantine approximations and continued fractions. If time permits, we will discuss invariant measures and understand conditional measures, which in my opinion are usually difficult to understand but become much more intuitive when considered as invariant measures. Depending the student’s interests, other related topics will be covered.
Suggested Book(s):
- A First Look at Rigorous Probability Theory, Rosenthal
- Probability and Measure, Billinglsey
- Ergodic Theory with a view towards Number Theory, Einsiedler and Ward; Techniques in Fractal Geometry, Falconer
- Introduction to the Modern Theory of Dynamical Systems, Katok
Graduate Mentor: Jiajie Zheng
An algebraic curve is algebraic variety of dimension one. An algebraic curve is the most frequently studied object in algebraic geometry. We will briefly introduce some basics of commutative algebra that we will use to discuss the theory of algebraic curves. Then we will talk about affine varieties. Our next goal will be to understand intersection numbers of algebraic curves. If time permits, we will cover the Riemann-Roch Theorem.
Suggested Book(s):
- Undergraduate Commutative Algebra, Reid; Algebraic Curves, Fulton
- Ideals, Varieties, and Algorithms, Cox, O'Shea, and Little
- A Royal Road to Algebraic Geometry, Holme
Graduate Mentor: Jiajie Zheng
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. It has crucial applications in optimization theory and mechanics, both classical and quantum. We first talk about arguably the most important theorems in calculus, the Inverse Function Theorem and its corollary the Implicit Function Theorem. Then we will talk about some classical variational problems, including Dido’s problem and Schrödinger’s equation. Then we will introduce optimal controls (boundary conditions are differential equations). Depending on the student’s interests, we can either cover rigorous theory of variations (through Banach space of functions and Fréchet derivatives, and generalized Inverse Function Theorem, of course), or weak and strong minima/sufficiencies.
Suggested Book(s):
- Calculus of Variations, MacCluer
- Functional Analysis, Calculus of Variations, and Optimal Control, Clarke
- Partial Differential Equations, Evans