Directed Reading Program Spring 2024

What is it? The Directed Reading Program (DRP) in the UW Madison Department of Mathematics pairs undergraduate students with graduate mentors for semester-long independent studies. During the semester, the student will work through a mathematical text and meet weekly to discuss it with their mentor. The original DRP was started by graduate students at the University of Chicago over a decade ago, and has had immense success. It has since spread to many other math departments who are members of the DRP Network.

Why be a student?

• Learn about exciting math from outside the mainstream curriculum!
• Prepare for future reading and research, including REUs!
• Meet other students interested in math!

Why be a mentor?

• Practice your mentorship skills!
• It strengthens our math community!
• Solidify your knowledge in a subject!

Current Organizers: Ivan Aidun, Allison Byars, Jake Fiedler, John Spoerl

Requirements

At least one hour per week spent in a mentor/mentee setting. Students spend about two hours a week on individual study, outside of mentor/mentee meetings. At the end, students give a 10-12 minute presentation at the end of the semester introducing their topic. This semester, it is scheduled for Wednesday, April 24th.

Applications

Check out our main page for examples of past projects.

Students: https://docs.google.com/forms/d/e/1FAIpQLSf2lm8Geuc6jwznBgGP5JjSJZuMITOw252e9qPOZCEQuFGIQw/viewform

Mentors: Applications are closed.

Questions?

Contact us at drp-organizers@g-groups.wisc.edu

Projects

Spring 2024 Projects

Title	Abstract	Required Background
Polynomial Methods	We will read Larry Guth’s book Polynomial Methods in Combinatorics and maybe some related short papers to understand how polynomials and their properties are applied to combinatorics, incidence geometry and harmonic analysis.	The material is accessible to students with linear algebra background. Corequisite of MATH 542 and 522 is recommended.
Algebraic Geometry	This project introduces scheme theory, a fundamental part of modern algebraic geometry, from a geometrically focused point of view. Topics to be covered will include affine schemes and their topology, an introduction to sheaves, structure sheaves, general schemes, and some examples. We will use Eisenbud and Harris' "The Geometry of Schemes."	The only strictly necessary requirement is familiarity with basic point set topology and commutative algebra at the level of a one-semester course. Rings, ideals, modules, localization, tensor products, etc. Basic algebraic geometry can be introduced at the beginning, but it would help to know about affine varieties, affine coordinate rings, and the correspondence between them.
Analysis	We will read Polynomial Methods in Combinatorics by Larry Guth. This covers a variety of applications of polynomials to different fields in mathematics, including combinatorics, analysis, and geometry. While I am most interested in analysis, we can choose what among the applications in the book interest you and focus on that.	Linear algebra, MATH 521-22.
Algebra/Probability	Free probability is a field which attempts to apply ideas from probability to more abstract settings, especially where the variables don't commute. In these settings we might not even be able to talk about 'probabilities', but thinking about expectations and distributions (suitably translated) can still tell us a lot. For example, it can tell us what the eigenvalues of a random matrix look like. Some funny things happen in the translation, like the role of the bell curve being taken by the semicircle (see Wigner's Semicircle Law for a related result). The material involves an interplay between things like operator algebras, combinatorics and probability, and would be interesting mostly to students who enjoy probability and didn't mind taking abstract algebra.	Students should be comfortable with linear algebra and have taken some abstract algebra (341, 541, and maybe 540 would be good). Some background in probability (not necessarily measure-theoretic) would help provide context. The material in the book doesn't require much heavy theory, so it should be possible to fill in any blanks along the way.
Real Analysis	Wavelets are a fun topic if you're interested in learning about a very useful application of both real analysis and linear algebra! They are used in lots of different areas of engineering (e.g. signal processing) and are interesting in their own right as a pure math tool! If you are interested in learning a more advanced topic of math, sign up for this DRP!	Real analysis (Math 521) and linear algebra (math 341 or 540) are prerequisites. If you have studied, Fourier series that's a plus but not a prerequisite! If you are motivated to learn something new, don't hesitate to sign up! We're all here to learn so questions are always and will always be welcome! I'm planning on using "An introduction to wavelets through linear algebra" by Michael Frazier.
History of Mathematics	We will read and discuss sections from John Stillwell's Mathematics and Its History. Which particular sections we read can be selected by any students in the group based on their interests. Our goal is to gain an appreciation for some of how mathematics became what it is today. What problems motivated its development, and how do our modern conceptions of things align (or not!) with what mathematicians were doing historically? What can we learn for doing math and other kinds of problem solving in the present?	We think interested students should have taken at least one proof-based math course. We think this project is unique in that it can accommodate students of widely varying pre-existing mathematical knowledge. Having little math knowledge means this project can give you perspective and intuition for math you will learn as you progress through the curriculum. Having a lot of math knowledge means this project can help you see the "big picture" of how facts you have learned connect to each other, sometimes in surprising ways.
Dynamical Systems (Reaction Networks)	Ever since Edward Lorenz found chaos emerging from his simple meteorological model, we have noticed that it is quite hard to try to understand the behavior of even very simple non-linear systems of differential equations. But these systems show up time and time again in nature, with a variety of very particular non-chaotic dynamics that we can make sense of. The study of Reaction Networks surprisingly manages to identify and classify these dynamics in systems that are very common in nature; from the enzymatic reactions on cells, to epidemiological models and even ecological population models. Using simple graph theory and linear algebra, we can uncover amazing results for the dynamics of these types of systems by abstracting some ideas. We will follow Martin Feinberg's book "Foundations of Chemical Reaction Network Theory". My goal in this DRP is to showcase this beautiful field of mathematics that lies in the intersection of both Pure and Applied Math, showing that these are more related that they usually seem, and how with abstraction and math we can uncover deeper truths about the structure of nature.	Students should be familiar with the language of elementary qualitative theory of ordinary differential equations (e. g., the meaning of asymptotic stability), good knowledge of modern linear algebra and calculus. For differential equations, Math 415 or Math 519 will suffice. For linear algebra, any course which dictates it is good enough.
Linear Representations of Finite Groups	We’ll be reading Serre’s Linear Representation of Finite Groups.	Abstract Algebra (Math 541) is expected. I expect the student to be somewhat independent due to time.
Geometric Measure Theory	Starting from the basics, the goal is to cover a solid amount of geometric measure theory, which is a field used to understand the geometry of complicated (but common in the real world) sets. We'll begin by discussing measures, and move into rectifiable curves/sets. We'll also talk about how projections affect the geometry of sets, in particular purely unrectifiable sets. Then, I hope to get into other topics like weak tangents and Plateau's problem generalized beyond smooth curves.	Students should have taken 521. Additional experience in proof-based math courses, especially analysis classes, will be very helpful. Students don't necessarily need to have seen measure theory, but it wouldn't hurt.
Abstract Algebra	We plan to follow the book Ideals, Varieties, and Algorithms by Cox-Little-O'Shea, roughly aiming to cover the first two chapters. This means we'll explore the relationship between algebraic objects (like polynomials) and geometric objects (like curves). We'll read about an important computational tool called Gröbner bases, which are used to find solutions to systems of polynomial equations (motivated by methods from linear algebra). This project, and the book we've chosen, will assume no prior knowledge of abstract algebra, and we'll learn any necessary concepts along the way! If you like a mix of theory and explicit examples, this project is for you!	This project is intended for students who have taken linear algebra and are interested in abstract algebra, but don't have much (or any) background in abstract algebra.

Presentation Schedule

Room 1 (VV 911)

Time	Speakers	Title
1:30-1:45	Dylan Wallace	The Frobenius Determinant and Linear Representations of Finite Groups
1:45-2:15	Yujun Che, Tianle Chen	Free Central Limit Theorem, Semicircular Element, and Dyck Paths
2:15-2:30	Tianze Huang	Arithmetic Scheme
2:30-2:45	Nithila Sivapunniyam	An Introduction to Gröbner Bases
3:45-4:00	Erkin Delic	Affine D-modules
4:00-4:30	Yingmo Zhang, Lejun Xu	Dynamic System in Reaction Networks
4:30-4:45	Ansh Aggarwal	A practical application of Real Analysis and Linear Algebra: Wavelets

Room 2 (VV B317)

Time	Speakers	Title
4:00-4:45	Braden Schleip, Benjamin Braiman, Oliver Jing	Geometric Measure Theory