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 firstname.lastname@example.org.
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.
Any sophomore, junior, or senior who has taken 15A (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.
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 2023 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 2023
Graduate Mentor: Alex Semendinger
Description: If you've generated images with Dall-E or Midjourney, or had a conversation with ChatGPT, then you have a sense of how impressive state-of-the art AI systems are now. But surprisingly, the most powerful AI systems today are built on relatively simple math: lots of linear algebra and a bit of calculus.
We will cover chapters 4-6 of the suggested text, including matrix decompositions, gradients of vector- and matrix-valued functions, multivariate Taylor series, and an overview of probability theory. Throughout, we'll make connections to machine learning techniques that apply each of these concepts, including linear regression, PCA, and especially deep neural networks.
- Mathematics for Machine Learning, Deisenroth, Faisal, and Ong (available online here)
Graduate mentor: Rocky Klein
Do you like beautiful designs with lots of symmetries? Whether it is brick walls, spirographs, honeycombs, rose windows in cathedrals, or intricate islamic patterns, chances are you have seen objects with many symmetries before. In this DRP we will be learning about the symmetries behind everyday objects. We will discuss tessellations of the plane and symmetries of three dimensional objects through the lens of group theory and low dimensional topology. We will start by characterizing the symmetries of basic objects and will then move on to classifying symmetries of the plane and of the sphere. There are exactly 17 symmetry types of planar tessellations, and this number seems to magically appear out of nowhere. We will follow the proof of “The Magic Theorem” which implies this fact. We will also discuss the Euler characteristic and orbifolds along the way. If time permits we will move on to finding generators for symmetry groups. The student may also select additional related topics by interest.
- The Symmetries of Things by Conway, Burgiel, and Goodman-Strauss.
Graduate mentor: Rebecca Rohrlich
Galois theory is the study of automorphisms of field extensions. This topic dates from the 19th century, when there was interest in formulas for roots of quintic (or higher degree) polynomial equations. Galois theory is now a fundamental tool in number theory for studying the arithmetic of number fields (finite extensions of the rationals). Our main goals for this reading course are to set up the necessary background, then learn and prove the Fundamental Theorem of Galois theory, and finally do LOTS of problems and applications of this theorem (they're really fun!). Time permitting, perhaps we can also talk about infinite Galois groups, or about open problems such as the inverse Galois problem. Only necessary background is to know what groups and fields are.
- Abstract Algebra (3rd ed.), Dummit and Foote
Graduate mentor: Ray Maresca
Does there exist a geometric realization of relative projectivity/injectivity in exceptional collections over type A quivers with any orientation? What about for linearly oriented quivers of type A -tilde?
Graduate mentor: Haochen Qiu
- Homotopy Type Theory by the Univalent Foundations Program
- 4-manifolds and Kirby Calculus by Gompf and Stipsicz