Directed Reading Program Fall 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

Applications

Requirements

At least one hour per week spent in a mentor/mentee setting. Students spend two to four 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, December 3rd.

Links to Apply

Mentors: Mentor Applications are closed for Fall 2025.

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

Students:

Application: Student Applications are closed for Fall 2025.

Questions?

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

Projects

The DRP offers two kinds of projects:

• "General area" projects. The mentor has proposed a general area that the project will be in, but the specifics of the project will be determined by the mentor and their students at the beginning of the semester. These projects will be flexible to meet the needs and level of students assigned to the project.

• "Specific book" projects. The mentor has proposed a specific book or other source to follow for the project.

Each project has a column noting which kind of project it is.

Title	Branch of Math	Type of project	Abstract	Required Background
Variational Calculus	Variational Calculus	General area	Calculus of variations is a branch of functional analysis that investigates the behavior of physical systems, trying to analytically discover functionals that optimize one (or more) physical properties under given constraints. We use the Euler-Lagrange equation, similar to the first derivative test, to help find functional solutions. The explorable topics include geometric optics, particle dynamics, inverse problems, numerical methods, eigenvalue/eigenfunction problems, elasticity, quantum mechanics, etc. The area will be determined by the student's interests.	Material in the textbook does not deal heavily with theory. A strong foundation in multivariable calculus and differential equations is needed. Knowledge in physics or coding is not required.
Mathematical Existential Crises	Logic, Philosophy of Math	General Area	What does it mean for a mathematical object to exist? Sometimes, in proving the existence of an object---such as a real number, a set of integers, or a graph coloring, each satisfying certain properties---it is possible to provide an explicit example. However, there are times when no such example can be given. In these cases, mathematicians often rely on "non-constructive" existential arguments, i.e., proofs of existence that do not provide a way to construct an example. In this project, we will explore several such non-constructive arguments across different areas of mathematics. Specific topics will be chosen according to the background and interests of the students. Potential directions include consequences of the Axiom of Choice, Erdős's probabilistic method, and applications of the Intermediate Value Theorem. Alternatively, one might take a historical perspective, examining the broader shift in mathematics from computational to conceptual styles of argumentation.	Students should have some prior experience with reading and writing proofs, at least at the level of an introductory proof-based course. Additional coursework in set theory, real analysis, or abstract algebra would be helpful but is not required.
Algebraic Geometry	Algebraic Geometry	General area	The goal of this DRP is to learn about some area of Algebraic Geometry. Depending on student interests and prior knowledge, the first few weeks may involve learning basic algebraic geometry before diving into more interesting areas. A couple possible topics I have in mind would be either to learn about algebraic curves and their moduli, or about tropical algebraic geometry. For algebraic curves, we would likely use either the book by Miranda or the book by Fulton, and the goal would be for students to learn key methods for studying algebraic curves, including understanding the Riemann–Roch theorem for curves. As a possible final project, they may investigate how Riemann–Roch leads to the fact that the moduli space of curves has dimension 3g−3. For tropical geometry the goal would be to learn about tropicalization of curves and then about tropical algebraic geometry which is algebraic geometry over the tropical semi-ring. We'd learn about what all of these words mean on the way. These are just some ideas and I'm open to many others.	Prerequisites: Algebra at the level of Math 541 (rings, ideals, homomorphisms, polynomial rings, etc) or equivalent. Helpful: to know some point-set topology at the level of 521.
Group Equivariant Convolutional Neural Networks	Algebra/Machine Learning	Specific book	Neural networks have proved excellent at identifying objects by recognizing their features. For example, a well trained network, when fed a pixilated image of the number 6, will be able to recognize the entire number based on its components, namely the circular base and curved line on top. In recent years, much work has been done in training networks to recognize such features after they have been altered slightly. As an example, can we train a network to recognize the number 4 when it has been drawn sideways? One approach which has proved incredibly effective is that of Group Equivariant Convolutional Neural Networks (GECNNs). For this reading group, we will work our way up from some basic definitions and concepts in neural networks. Then, we will take a look at convolutional neural networks, applied to the number recognition task. Lastly, we will work with GECNNs, applied to the rotated number recognition task. If we have time, I would love to look at steerable CNNs, which are an application of representation theory. I plan to use reading material from Ian Goodfellow’s Deep Learning textbook, some of 3Blue1Brown’s videos, Cohen and Welling’s paper on GECNNs, and some open courseware material from the University of Amsterdam.	Students interested in this reading program should have familiarity with calculus, linear algebra, and some experience with coding in python. In practice, the group theory is minimal. So, familiarity with algebra is not necessary.
Quiver Representations	Algebra	Specific book	Background in algebra is hoped for. I'm really hoping to use Kirilov's book on Quiver Representations and Quiver Varieties. Probably on the headier side of the list of topics. Depending on the student, I might switch to an easier book on Quiver Representations.	Algebra Background
Category Theory and Algebraic K-theory	Algebra	Specific book	Algebraic K-theory is one of the most important methods in modern mathematics. It plays a vital role in number theory, algebraic geometry, and homotopy theory. As a mathematical object, it heavily depends on the theory of categories. In this language, many early mathematical concepts can be organized in a coherent way. We will first do some reading on category theory, understanding morphisms, Yoneda lemma, colimits, and adjunctions. A good reference is Mac Lane’s book Categories for the Working Mathematician. Afterwards, we will discuss the construction of algebraic K-theory.	Familiarity with commutative algebra and module theory.
Uncertainty Principles and Signal Recovery	Discrete Fourier Analysis	Specific book	This project will focus on recent, interesting applications of Fourier analysis to some problems in signal processing. We will read and try to understand the entirety of the recent paper "Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery" by Alex Iosevich and Azita Mayeli (https://arxiv.org/pdf/2311.04331). If we get through that, I have other recent papers in mind to look at. There has been a lot of recent progress in this area.	Some familiarity with Fourier series (Math 629 or higher) will be useful. Having some coding experience will also be nice. I expect some basic exposure to analysis (Math 521/522).
The geometry of fractal sets	Analysis	Specific book	This semester, we'll be reading Falconer's Geometry of Fractal Sets. We'll begin with an introduction to measure and dimension, then study density and some of the structural properties of fractal sets. Time permitting, we'll also read about projections and Kakeya sets.	Ideally, students will have completed 521
Understanding and applying Neural ODEs	Applied and computational math	Specific book	This project will introduce students to Neural Ordinary Differential Equations (Neural ODEs)—a modern framework at the intersection of differential equations, machine learning, and scientific modeling. First, students will build a foundation by reviewing concepts from ODEs, numerical methods, optimization and basic machine learning (depending on the students strengths and weaknesses). Then, the project will emphasize both theory and practice: students will study the mathematical formulation of Neural ODEs, implement computational models, and apply them to problems drawn from physics and applied mathematics.	ODEs, numerical analysis from math and basic coding skills (python) from computer science.
The Mathematics of Modeling Thin Sheets	Computational mathematics and modeling	Specific book	This project will be an in-depth exploration of modeling thin elastic sheets. Crumple theory studies the crumpling and wrinkling of sheets as a proxy for damage accumulation in mechanical systems. The theory is applicable in fields such as animation and materials science. Accurately representing the continuous sheet using some discrete mesh, in order to properly simulate crumpling, is a very important piece of this theory. We'll investigate a mass-spring triangular mesh of the sheet and answer the question, ""what does it mean for this model to be 'good'?"" Concepts of accuracy and convergence are fundamental in computational mathematics. We will primarily follow the paper by Leembruggen et al., "Computational model of twisted elastic ribbons" which can be found at https://doi.org/10.1103/PhysRevE.108.015003, and supplement background information as needed.	Students should have interest in the mathematical modeling of physical systems. Ideally, one should be familiar with introductory numerical analysis and physics concepts seen in classical mechanics courses. This is a niche and rich topic of study and is very interesting on its own, but students could approach this as a means to strengthen their understanding of mathematical modeling, physics, and computational mathematics in general. This project may look intimidating, but we'll plan to work at a manageable pace and determine a starting point based on the group's backgrounds.
Stability in geometric invariant theory	Geometric invariant theory.	Specific book	In general, group actions on algebro-geometric objects do not admit obvious quotients. Mumford's geometric invariant theory provides an efficient and natural method for constructing good quotients of varieties subject to stability constraints, and it proved to an extremely successful and useful theory. We will read Newstead's short text, Introduction to Moduli Problems and Orbit Spaces. Beginning with constructions for affine varieties using Nagata's theorem, we will study the geometry of orbits, examining stability and thereby establishing the existence of good (and in some cases geometric) quotients of projective varieties. The ultimate goal is to understand the Hilbert-Mumford criterion and to work out the stability of objects in various moduli problems. Well-versed participants may also explore selected topics in Mumford's GIT book if time permits. This project can be adapted flexibly depending on the participants. Our main reference will be Introduction to Moduli Problems and Orbit Spaces.	Some familiarity with basic facts about classical varieties over the complex numbers is required, though a background in general commutative algebra is not. A good understanding of polynomial rings will be sufficient. We will treat algebraic groups through explicit examples, so again no general theory is involved. The language of schemes will not be used in this project, though participants interested in schemes may certainly choose to use them. At some point we will encounter algebraic curves and vector bundles, but no prior knowledge is required.