Schedule
Kick-off event: TBA
Mid-semester check-in: TBA
End-of-semester presentations TBA
Fall 2026 Projects
Beginner Project
Mentor: Summer Haag
Mentees: TBD
Title: Introduction to Visualizing Mathematics
Abstract: For this project, we will work through chapters of Indra's Pearls: The Vision of Felix Klein, which explores patterns through iterated möbius transformations and their connections with symmetry, geometry, number theory, abstract algebra, and computer graphics. This is an interactive book so we will be doing exercises and visualizations through coding together.
While reading this book, there are many directions we can go, for example focusing on groups, algorithms, fractals like Apollonian circle packings (my favorite), geometry, or the connections between any of these. Interest and time will dictate how we travel around this material and I hope we can make some cool and interesting pictures together.
This is totally introductory but can be intermediate/advanced by just skipping certain prerequisites and getting to harder questions faster. Classes that could be helpful but are not strictly necessary are abstract algebra, linear algebra, complex analysis, discrete math, and number theory.
Prerequisites: This book is completely introductory and so no prerequisites are needed but experience with abstract algebra or complex numbers will be helpful.
Beginner/Intermediate Project
Mentor: Anoushka Nerella
Mentees: TBD
Title: Convex Optimization
Abstract: In this reading group, we aim to learn the theory of convex optimization and some applications of it. We will follow the textbook “Convex Optimization” by Stephen Boyd and Lieven Vandenberghe. This book contains a wide range of examples which should help us in understanding the theory through active problem-solving. The reading plan will focus on selected topics from Part I of the book, including convex sets, convex functions, and basic optimization problems. The goal of this theoretical component is to develop a solid conceptual and geometric understanding of convexity and optimization techniques. Building on this foundation, we will then study selected material from Part II, particularly the chapter on Approximations and Fitting, which introduces important applications such as least squares and data fitting methods.
The primary prerequisite for this project is a working knowledge of linear algebra, including vector spaces, matrices, eigenvalues, and inner products. Some familiarity with mathematical reasoning and proof-writing is helpful but not required; part of the goal of the reading group is to help participants develop these skills.
As a summer reading prior to the program, I suggest participants go through chapter 0 (Introduction) of the book so that everyone is on the same page with Linear Algebra. In case of any confusion with the prerequisites, I request you to contact me. I can help fill any gaps you may have with respect to Linear Algebra. The book is available for free on the Stanford website and I can send a PDF copy upon request.
Prerequisites: Linear algebra and some familiarity with proofs.
Intermediate Project
Mentor: Colin Jackson
Mentees: TBD
Title: Irrational Numbers
Abstract: The first numbers we learn about are the integers, followed closely by the rational numbers, which are just ratios of integers. But what else is there? It turns out that there are plenty more numbers out there, which are called the irrational numbers. The goal of this project will be to study irrational numbers and their relationship to the rational numbers. For example, how do we know which numbers are irrational? How closely can we approximate an irrational number by a rational number? Are there more rational numbers or irrational numbers? After this, we will also look at transcendental numbers, such as e and π. While every transcendental number is irrational, not every irrational number is transcendental. It turns out to be much harder to prove that a number is transcendental.
We will mostly use Ivan Niven's "Numbers: Rational and Irrational," which you can access a PDF of through the CU library. Another option is David Angell's "Irrational and Transcendence in Number Theory."
Required background is Introduction to Discrete Math and Calculus 2, just to have a base knowledge of proof writing and comfortability with integrals. The level of this project can be adjusted according to interest from students, so please reach out if you are interested (even if you don't necessarily fit the background above)!
Prerequisites: Introduction to Discrete Math and Calculus 2.
Intermediate Project
Mentor: Basia Klos
Mentees: TBD
Title: Exploring Lie Algebras
Abstract: A Lie (pronounced LEE) algebra is a vector space equipped with a multiplication that is in general neither commutative nor associative. These objects show up in many areas of mathematics, including differential geometry, algebraic topology, and even mathematical physics. In this DRP, we will start by getting acquainted with the definition of a Lie algebra and some friendly examples. We'll then move to understanding deeper results, such as (depending on student interest) representation theory of Lie algebras, the classification of finite-dimensional semisimple Lie algebras over the complex numbers, or infinite-dimensional Lie algebras and how they give rise to vertex algebras. We will follow the introductory Lie algebra textbooks of Erdmann-Wildon and Humphreys. If you enjoyed linear algebra and want to apply that knowledge to higher math, then this DRP is for you!
Prerequisites: Linear algebra and a familiarity with proofs (discrete math).
Intermediate/Advanced Project
Mentor: Courtney Hauf
Mentees: TBD
Title: Equivariant Topology
Abstract: In this project, we will learn the fundamentals of equivariant topology. Topology is the study of spaces and continuous maps, and equivariant topology is the study of spaces with a group action and equivariant maps, those continuous maps that play nicely with the group action. The way to encode this additional equivariant structure varies throughout different branches of mathematics, and we will work to identify the benefits and drawbacks to a few of these approaches.
Depending on the backgrounds of mentees, the project will start by going over the basics of group theory (groups, subgroups, homomorphisms) and point-set topology (spaces, subspaces, maps, paths) drawing connections with the topology inherently used throughout calculus. From there, we will introduce spaces with a group action and begin to explore some models of encoding this data into the space structure. We’ll look to see how, and if, fundamental results of ordinary topology lift to the equivariant world. All references will be provided for students, and will consist of excerpts from books, notes, and expository papers, some useful videos, and some personal notes.
Prerequisites: Students should have taken a proof-based course. This project can be tailored to students’ specific interests and background, as final topics will lean into their preference of structures and interest in fundamental results. Students having taken Algebra 1 and Topology can expect to dive deeper into the equivariant structures and fundamental results, but such courses are not necessary for participation.
Advanced Project
Mentor: Jon Kim
Mentees: TBD
Title: Ideals, Varieties, and Algorithms
Abstract: Algebraic geometry at its core is the study of solutions to a set of polynomials. While this might not seem interesting on the surface, these solution sets host a vast array of rich geometric properties that have strong connections to the algebraic properties of the polynomials. In this project, we will read through the book Ideals, Varieties, and Algorithms by Cox, Little, O'Shea (available for free on Springer). This is a great introductory text in commutative algebra and algebraic geometry with a focus on the computational aspects. There are two main goals we can explore:
1. Get familiar with Macualay2: a computational programming language for algebraic geometry.
2. Get to the Nullstellensatz theorems: foundational theorems in algebraic geometry.
Prerequisites: Abstract algebra (ring theory).