Directed Reading Program Spring 2025

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, Ari Davidovsky, 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, presentations will be Wednesday, April 23rd.

Applications

You can find examples of past projects from the DRP main page.

Students:

Application available here: https://forms.gle/YRPTFoaoFDQ4aMX77. The application will close on Thursday January 30th at 11:59pm.

Questions?

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

Projects

Title	Branch of Math	Abstract	Required Background
Score-Based Diffusion Models	Applied Mathematics	This program introduces advanced undergraduates to the foundational concepts of Stochastic Differential Equations (SDEs) and their application in modern machine learning through score-based diffusion models. Drawing from Lawrence Evans’ An Introduction to Stochastic Differential Equations, which simplifies the graduate-level subject for strong undergraduates by focusing on intuition and omitting measure-theoretic details, participants will explore topics such as Brownian motion, random calculus, and their connections to generative modeling. This program bridges theoretical mathematics and cutting-edge AI, offering students a unique opportunity to engage with stochastic processes and their transformative impact on machine learning.	Probability, calculus, and familiarity with differential equations and linear algebra.
PDE-constrained optimization for physic systems	Applied Mathematics	Physical systems are typically described by partial differential equations (PDEs), and PDE-constrained optimization can be a useful tool for controlling the instabilities that arise in these systems. We will begin with an introduction to inverse problems in physical settings and the corresponding PDE-constrained optimization problems. Next, we will explore the ill-conditioning of inverse elliptic equations and the inverse heat conduction problem, and examine some numerical techniques to address their ill-posedness. If time permits, we may explore more complex physical systems of interest and apply PDE-constrained optimization tools to analyze their instability and control techniques. The expected outcome is to understand the PDE formulation of certain physical systems and interpret the ill-conditioning of their inverse problems.	320, preferably programming.
Graph Theory and Applications	Combinatorics	This topic will explore graph theory from a proofs based perspective with a few algorithms on graphs (beyond Dijkstra's). The study will follow Douglas West's: Introduction to Graph Theory very closely. Ideally, the study would cover some preliminary concepts of the following: 1. Basic concepts 2. Connectivity 3. Bipartite (Non-bipartite if time permits) matchings 4. Graph colouring Algorithms such as Edmonds-Karp for network flow would be included as it covers both parts 2 and 3.	The student, ideally, must have at least one proof based math course under their belt. An introduction to graph theory either by the Discrete Mathematics course or an Intro to Algorithms course by the CS department is also desirable. The student must also understand Big O notation to understand efficiency of certain algorithms. The student should be willing to work out proofs for certain problems on their own in order to deeply understand the underlying concepts.
Quadratic Number Fields	Algebra/Number Theory	We will study quadratic number fields and their rings of integers. This provides an introduction to the field of algebraic number theory, and can be used to understand (among other results) solutions to Pell's equation, a proof of Fermat's last theorem for the case n=3, and a proof of quadratic reciprocity.	They should have taken math 541 or an equivalent course in abstract algebra. At points it may be helpful to either be currently enrolled or have taken math 542.
Character theory of finite groups	Algebra	This DRP will serve as an introduction to representation theory of finite groups. First we will define representations and group algebras, and related constructions. Then we will study characters and their main properties. Along the way we will cover many essential results, like Maschke's theorem, Schur's Lemma and the Orthogonality Relations. Depending on your interests and if time permits, we may see additional topics, such as applications to group theory, rationality questions, or representations of symmetric groups. We will likely use James and Liebeck's book "Representations and Characters of Groups", at least in part.	Some familiarity with basic concepts in abstract algebra is required.
Fractal Geometry	Analysis	We will read the book 'Measure, Topology, and Fractal Geometry' by Gerald Edgar together. Roughly speaking, we will study different ways to define the dimension of a set, including several topological dimensions and Hausdorff dimension and the properties of the above dimensions. We will also study sets with dimension not equal to a natural number, the most famous one being the Canter middle third set.	The minimum prerequisites are those given in the book by Edgar: 1)Experience in reading and writing mathematical proofs, 2) Basic abstract set theory (finite vs. infinite sets, countable vs. uncountable sets, etc.), 3)Calculus. It would be great to be familiar with topics covered in Math 521, but it's not necessary since all those are also covered in the book by Edgar.
Topology from the Differentiable Viewpoint	Geometry/Topology	We will read Milnor’s “Topology from the Differentiable Viewpoint”. This book uses multivariable calculus as a way to study topology. This will give us just one of many perspectives on the study of special objects called manifolds. Manifolds are nice shapes that look like R^n when looked at up close. Examples include the circle, the sphere, the torus, and R^n itself. Along the way we will see how to take derivatives of functions between manifolds, classify all of the functions between certain manifolds, find an alternative way of computing the Euler characteristic, and more if time permits.	Some of the words in the abstract may sound scary, but no prior topology experience is required. Students should have familiarity with multivariable calculus, linear algebra, and at least one proof based math course.
Kolmogorov Complexity	Logic	Consider the two binary strings 00000000000000000000 and 10111001001101000111. Intuitively, the second string seems "more random", but making this intuition mathematically precise is challenging (for instance, both strings are equally likely as the result of 20 fair coin tosses). Kolmogorov complexity is way of quantifying the information content of strings and other objects. To study this notion, this DRP will start with an introduction to computability theory. Then, we will use Turing machines to define Kolmogorov complexity and several of its variants. We will investigate a number of important inequalities and identities involving Kolmogorov complexity, especially symmetry of information. Finally, time permitting, we will discuss several applications.	The completion of at least one proof-based math class would be very helpful.
Math of voting and elections	Discrete math	Most elections in the United States use a voting method called plurality, but there are many other possible voting methods when we're voting on more than two candidates. Voting theory is an area of math which studies different voting methods from a mathematical point of view. A well-known theorem in this area, called Arrow's impossibility theorem, says that if we try to impose a few reasonable-sounding restrictions on our voting methods, any method you come up with will violate at least one of them. In other words, there is no "perfect" voting method when a group of people is deciding between more than two options. We will explore different voting methods and their pros and cons, and we will work to understand a proof of Arrow's theorem. Depending on interest, we may read about related topics including apportionment, fair division problems, or redistricting.	Familiarity with the material from Math 240 (mathematical proofs and definitions of sets, functions, etc.) would be helpful but is not required! This area offers an opportunity to use mathematical reasoning without much technical background.
Hyperbolic Geometry	Geometry	We will learn about two-dimensional hyperbolic geometry. We will start with the upper half plane model of the hyperbolic plane and learn how to work with its isometry group. Then, we will see how to do computations in the hyperbolic plane, including calculating length and distances. We will then explore other models of the hyperbolic plane. Depending on time, we may explore how hyperbolic geometry fits into modern mathematics, including connections to dynamics, topology, and algebraic geometry.	The main prerequisite is a proof-based math class, ideally a 500 level class like 521 or 541. Additionally, the students should have seen the material of Calc 3 (234), and, since we will be working with matrices, some linear algebra class. Abstract algebra is helpful but not required.
Mathematical and Algorithmic Applications of Linear Algebra	Linear algebra and combinatorics	We will read "Linear algebra methods in combinatorics" by L. Babai and P. Frankl. This is a great self-contained book! It requires some past practice with proof-writing but not much more than that. We will explore how one can use seemingly elementary tools from linear algebra to obtain interesting results in combinatorics. I have also selected a few bite-sized, miniature proofs in another book: "Mathematical and Algorithmic Applications of Linear Algebra". Our goal is to gain familiarity with intriguing ideas and proofs which rely on seemingly simple linear algebra tools! If you're interested, sign up!	The requirements are linear algebra, discrete math and a real analysis course. A semester or two of practice writing proofs is a must but you don't need much more than that! The resources I have collected are self-contained and I'd be happy to answer any questions you have along the way.
Applications of Dynamics and Chaos	Dynamical Systems	Dynamical systems study what happens to a space when we have a map or process act on a space repeatedly. We call a system chaotic if a small change in the initial conditions of our system greatly alters our system. Through this simple setup we can create some beautiful mathematics and stunning visualizations. These ideas have a wide variety of applications in physics, biology, weather prediction, economics, and other areas. In this DRP we will study these systems as well as some of their applications.	Students should have completed the calculus sequence (through 234). An introductory physics sequence will be helpful but not strictly required. Additionally linear algebra and a course in differential equations may be useful.

Presentation Schedule

Presentations will be held Wednesday, April 23rd. Presentation schedule will be posted here once it is determined.

Time

Speakers

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Mentor