Directed Reading Program in Mathematics@UF
Welcome to the Directed Reading Program (DRP) in Mathematics at the University of Florida! This new initiative partners/groups undergraduate students with graduate mentors for semester-long independent reading projects in mathematics. Through this program, students explore topics outside the standard curriculum, build mentorships, and deepen their mathematical understanding.
What is the DRP?
The Directed Reading Program is a mentoring initiative designed to give motivated undergraduates an opportunity to explore advanced mathematics topics under the guidance of a graduate student mentor. Over the course of a semester, each student-mentor pair/group meets regularly to work through a math paper or book chapter(s), culminating in a brief presentation at the end of the term.
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.
How It Works
- Pairings/Groups: Undergraduate students submit a ranked choice of readings, and we connect them with the graduate students.
- Meetings: Pairings/groups meet weekly (typically for 1 hour) to discuss the material, ask questions, and work through challenging ideas.
- Final Presentation: At the end of the term, students give short (10–15 minute) presentations on their projects in a friendly, supportive setting.
We will do our best to match every applicant with an appropriate mentor though spots may be limited.
Goals of the Program
- Support undergraduates in exploring math beyond coursework.
- Provide mentorship and foster a collaborative learning environment.
- Prepare students for research opportunities and graduate school.
- Provide mathematics graduate students an opportunity for mentoring experience in their area of interest, as well as deepen their own understanding of their research area.
- Strengthen the mathematical community at UF.
Who Can Participate?
Undergraduate Students
We welcome applications from all UF undergraduate majors although priority will be given to mathematics majors.
Graduate Mentors
Graduate students in mathematics can apply to serve as mentors. Mentors choose reading materials, meet weekly with their mentee(s), and guide them through the reading. Appropriate reading materials might include Math Monthly or other similarly accessible articles, foundational or introductory articles in the graduate student’s research field, or even book chapters. This is a great opportunity to build mentoring experience in a meaningful way.
Timeline – Spring 27
- Undergraduate Mentee Applications: will open by November 13, 2026 and close November 30, 2026, at 2pm.
- Pairs/Groups Announced: by December 18, 2026.
- Reading Period: January 11-April, 21, 2027.
- Final Presentations: Participants will have the opportunity to present either a short talk or a poster at the 2027 Undergraduate Mathematics Symposium, on Friday, April 23 (Room and Time: TBA).
How to Apply
Undergraduates: Complete this form https://forms.gle/xrqPZtr6EAyLYwuH9 by November 30, 2026, at 2pm.
Note: To access the form, you must login to Google forms with your @ufl.edu email: https://cloud.it.ufl.edu/collaboration-tools/g-suite/
Graduate Mentors: check your email!
Spring 2027 Readings
Please note that students will only be assigned to projects for which they meet the prerequisites.
1. Advanced Linear Algebra by Steven Roman
Reading material: Chapters 3, 9, 12, and 13. If possible, I would also like to cover either Chapter 11 or Chapter 14 if there is time left over from the other content.
Description: I want to motivate the study of linear algebra beyond what the students will have seen in proof-based undergraduate linear algebra, particularly in the infinite-dimensional setting. The goal of the course will be to introduce more advanced linear algebra concepts and some basic analytic concepts, with the goal of introducing Hilbert spaces and proving the Riesz Representation Theorem for Hilbert spaces. Chapter 3 covers quotient spaces, dual spaces, and adjoints (~20 pages). Chapter 9 covers real and complex inner product spaces, isometries, orthogonality, and the Riesz Representation Theorem for finite-dimensional vector spaces (~18 pages). Chapter 12 is about metric spaces and discusses open and closed sets, convergence, dense subsets, continuity, and completeness (~20 pages). Finally, Chapter 13 covers Hilbert spaces and covers series, Hilbert bases, Fourier expansions, a characterization of separable Hilbert spaces, and the Riesz Representation Theorem (~25 pages). [Total ~83 pages of content for the main focus of the DRP]
If there is still time, I would like to either cover select sections from Chapter 11 of Roman, which covers bilinear, quadratic, and symplectic forms and their applications to geometry on vector spaces, ending with the Witt theorems (< 40 pages) or cover select sections from Chapter 14 of Roman which is about tensor products and algebras of vector spaces. The primary focus of the course will be about infinite-dimensional vector spaces and the motivation of the Riesz Representation Theorem.
I chose Roman’s book because he very carefully motivates the Riesz Representation Theorem across the entire book, beginning in chapter 9. Roman wrote this book targeting advanced undergraduate students and graduate students, so I think it is appropriately scoped for a student who has already taken a course in linear algebra. Additionally, it is free online, which is a plus.
Prerequisites: MAS 4105, MAC 2313
Recommended background: MAA 4211 is helpful, but not necessary.
2. Geometry of Fermat’s sum of squares by Greg McShane and Vlad Sergiescu
Description: This reading focuses on the connection between Fermat’s sum of squares theorem, a classical result in number theory, and hyperbolic geometry.
Prerequisites: MAA 4226, MAA 4402
Recommended background: Some familiarity with the special linear group is helpful
3. Orthogonal Polynomials on the Unit Circle, Part 1 by Barry Simon (AMS Colloquium Publications)
Reading material: Selected topics from Chapters 1 and 2.
Description: In analysis, Szegő limit theorems are beautiful results describing asymptotics of Toeplitz determinants. In the suggested material, these theorems have been viewed through the lens of orthogonal polynomials on the unit circle (OPUC). Using the nice recurrence relations that OPUC satisfy, one can prove a long list of interesting equalities. We will try to cover some topics from the first two chapters of the book, which includes beautiful proofs of the Szegő theorem along with historical remarks. A basic course in complex analysis and linear algebra is enough to understand the material.
Prerequisites: MAS 4105 Linear Algebra 1, MAA 4402 Functions of a Complex Variable
4. Markov Chains and Mixing Times by David Levin, Yuval Peres, and Elizabeth Wilmer
Reading material: Chapter 1 and topics from Chapter 2.
Description: This reading project introduces students to finite Markov chains. Markov chains provide a framework for understanding random processes. Topics include transition matrices, stationary distributions, and random walks. We will also look at applications such as gambler’s ruin and the coupon collector problem. Depending on the students’ interests, some computations and simulations could be explored. The project requires introductory probability and linear algebra.
The first two chapters provide an accessible introduction to the topic, beginning with elementary concepts. They also provide interesting applications and opportunities for exploring topics through computations.
Prerequisites: STA 4321 (or equivalent introductory probability experience), MAS 3114 OR MAS 4105 (or equivalent linear algebra experience), MHF 3202 (or experience writing proofs) preferred
5. Dirichlet series and L-functions in number theory
Reading material: Tom M. Apostol, Introduction to Analytic Number Theory, Chapter 11.
Description: This project explores how Dirichlet series and Euler products connect analysis with arithmetic. Students will study convergence properties, multiplication of Dirichlet series, and the role of multiplicative coefficients, beginning with the Riemann zeta function and progressing to Dirichlet (L)-functions. And we’ll see some beautiful relations with analytic and algebraic information via these functions. All of this will be covered from Chapter 11 of the mentioned book. Towards the end, we will investigate how similar ideas lead to (L)-functions associated with modular forms and elliptic curves, highlighting connections between Fourier coefficients, prime numbers, and arithmetic properties of elliptic curves without requiring an advanced background in modular forms and elliptic curves.
Prerequisites: MAA 4211, MAA 4402
6. Graph Spectra and Continuous Quantum Walks by Gabriel Coutinho and Chris Godsil
Description: This book is an interesting introduction to quantum walks on graphs, a relatively recent research subject that mainly uses ideas from graph theory and linear algebra, with applications to quantum information and quantum computing. This reading group will provide an opportunity to learn more advanced linear algebra than what students may be used to encountering. The book is also very adaptable, as we can make the reading more computational or applied depending on the interests of the students.
Prerequisites: MAS 3114 Computational Linear Algebra
Recommended background: MAS 4105 Linear Algebra 1, programming
7. A Visual Introduction to Differential Forms and Calculus on Manifolds, by Jon Pierre Fortney
Description: Differential forms provide a unifying language for using calculus to study the geometry of higher-dimensional spaces called manifolds. This reading group will explore the connections between multivariable calculus, linear algebra, and differential geometry, emphasizing visual intuition and computational fluency with differential forms. We will investigate how familiar concepts from vector calculus, including integration, vector fields, and the fundamental theorem of calculus, can be extended to curved spaces. Topics will include tangent spaces, differential forms, the wedge product, exterior derivatives, and, if time permits, integration on manifolds, culminating in a study of the generalized Stokes’ theorem.
Prerequisites: MAC 2313 and MAS 4105.
8. Notes on Basic 3-Manifold Topology by Allen Hatcher
Reading material: Chapter 1.
Description: The goal of this course is to understand the prime decomposition and, time permitting, the JSJ decomposition of a compact 3-manifold. These decompositions give ways of cutting up general 3-dimensional spaces into pieces which are easier to understand. The ideas in this course will give students the foundation to appreciate the classification of 3-manifolds, which is one of the greatest triumphs of modern geometry.
Prerequisites: MTG 4303
Recommended background: Covering space theory
DRP Archives
Questions?
Email Konstantina Christodoulopoulou (kchristod@ufl.edu) or Sara Pollock (s.pollock@ufl.edu).