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Department of Mathematics

DRP@UF

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.  Students are  enrolled in 1 credit hour of MAT4905 (Individual Work).
  • 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 2026

  • Graduate Mentor Applications close: Friday, October 24, 2025.
  • Undergraduate Mentee Applications: Close on  November 21, 2025  at 2pm.
  • Pairs/Groups Announced: by December 8, 2025.
  • Reading Period:  January 12-April 22, 2026.
  • Final Presentations:  End of  the spring semester.

How to Apply

Undergraduates: Complete this form by November 21:   https://forms.gle/LYuuyrhYTRmddXG1A 

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 26 Readings

Please note that students will only be assigned to projects for which they meet the prerequisites.

 1. Homotopic distance between maps by E. Macías-Virgós & D. Mosquera Lois

  • Description: Given two continuous maps between topological spaces, how does one go about quantifying how “far” apart the two maps are? One way to do this is to study the “homotopic distance” between the two maps in terms of the subsets of the two topological spaces involved. This material is an introduction to this line of ideas. From this material, folks will be able to understand how the homotopic distance between maps behaves as an invariant in a variety of situations and how its properties are understood using standard tools from algebra and topology. A remarkable feature of the notion of homotopic distance is that it recovers the notion of topological complexity of a topological space. The latter is a topological invariant used in the study of the motion planning problem from the field of robotics.
  • Prerequisites: (MAS 4301 or MAS 4105) and  MHF 3202, and (MAA 4102 or MAA4211).
  • Recommended background: MTG 4302

2. Avoiding monochromatic solutions to 3-term equations by Kevin P. Costello & Gabriel Elvin

  • Description: This paper provides a relatively gentle introduction to arithmetic Ramsey theory. It combines ideas from additive combinatorics and Fourier analysis to study monochromatic solutions to linear equations. Understanding the paper requires solid proficiency in undergraduate analysis and some familiarity with undergraduate algebra. No prior background in combinatorics, number theory, or Fourier analysis is necessary; these can be supplemented as needed throughout the semester through additional resources. Many open questions remain in this line of research, so this reading is suitable for students looking for combinatorial research topics.
  • Prerequisites: MAA 4211 and MAS 4301.

3.  Finite Element Method (Lectures by Xiaoming He)

  • Description: As an important tool in numerical methods, the finite element method (FEM) has drawn significant attention for its power in solving complex physical and engineering problems. Despite its widespread use, translating FEM concepts into code can be intimidating, especially for those new to the field. In this reading group, our goal is not to focus on high-level mathematical theory, but to provide an accessible and intuitive introduction to FEM. We aim to present the key ideas from the ground up, making it possible for even freshmen to grasp the core concepts and appreciate how FEM works. Through simple explanations and hands-on coding examples, we hope to offer a fresh perspective and make the learning process enjoyable and engaging.
  • Prerequisites: MAD 2502 and (MAS 3114 or MAS4105).
  • Recommended background: Matlab or Python.

4. Chap. 9-12 from the book Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz

  • Description: Almost all phenomena in biology, chemistry, or physics are complex, non-linear systems that evolve over time. If you’re interested in understanding and explaining such phenomena for intellectual curiosity or innovative purposes, you may find it beneficial to learn more about these fascinating mathematical concepts. These fundamental concepts will equip you to comprehend, explain, and challenge various natural phenomena that follow non-linear patterns. The topics you’ll study include: Chaos Theory and its application in sending secret messages, Strange Attractors, the Lorenz Map, the Logistic Map, the Henon Map, Fractals, stability, fixed points, the Cantor Set, and orbits. This practical mathematical knowledge is incredibly valuable to a mathematician, offering a strong foundation for exploring and understanding the complexity of the natural world.
  • Prerequisites: MAP 2302 and (MAS 3114 or MAS 4105).

5. Ultrafilters Throughout Mathematics by Isaac Goldbring

  • Description: Ultrafilters are a mathematical object traditionally used in mathematical logic, but have been applied in a variety of mathematical disciplines such as topology, combinatorics, algebra, analysis, and voting theory. The first chapter of the book provides an introduction to ultrafilters, and from there selected chapters can be chosen to match with the reader’s interests in any of the above areas.
  • Prerequisites: MAS 4105.
  • Recommended background: MAA 4102 or 4211, MAS 4301, MTG 4302 (all are recommended; strongly recommended to have at least one)

6. How to mathematically optimize drug regimens using optimal control by Helen Moore

  • Description: Optimal control is a powerful mathematical tool that can be used in medicine to optimize drug regimens (make treatments more effective while also minimizing side effects!) This article is written for applied mathematicians who have not used optimal control before. It is written in a very accessible way in that “basic” concepts, such as partial derivatives, have reminders and definitions, making it an ideal paper to work through with undergraduates who may not have a lot of mathematical experience. Several applications are discussed which would be great for the undergraduates to work through and present.
  • Prerequisites: MAC2311 required, MAC2313 and MAP2302 would be helpful but are not necessary. MAP4484 would be ideal, but not necessary.
  • Recommended background: Beginners are very welcome. Students studying math or (pre-)medicine are especially encouraged. All you need is an interest in how math can be used in medicine. The rest you can learn!

7. Principles of Mathematical Modeling (Second Edition) – by Clive L. Dym [ Chapter 1: What is mathematical modeling ?, Chapter 2: Dimensional Analysis, Chapter 3: Scale, Chapter 4: Approximating and validating models, Chapter 5: Exponential Growth and Decay, Chapter 6: Traffic Flow models (if time permits)]

  • Description: This reading aims to focus on key concepts and techniques in mathematical modeling and their applications. It will give readers the opportunity to strengthen foundational modeling skills and apply them to real-world problems by developing and analyzing mathematical models.
  • Prerequisites: MAP 2302.

8. Introductory Functional Analysis with Applications, by Erwin Kreyszig (selected sections of Ch.2 and Ch.3, more if time permits)

  • Description: This textbook gives an accessible and detailed treatment of several fundamental results about Banach and Hilbert spaces. It does not assume familiarity with measure theory, making it accessible for students who have not taken a graduate-level analysis course. Bounded linear operators on Banach and Hilbert spaces are one of (if not the most) important objects in analysis, so we will spend a good amount of time learning about them. The text covers enough material for me to talk about some of the basic ideas and results in dilation theory, which is related to my current research.
  • Prerequisites: MAS4105 (some text sections cover this material but they will be skipped), MAA4211/4212 (metric spaces; Ch.1 of the text covers the relevant material)
  • Recommended background: Familiarity with basic analysis.

9. Chapter 9 (Zero density estimates) from The Riemann Zeta-Function: Theory and Applications, by  Ivic, Aleksandar.  N.p.: Dover Publications, Incorporated, 2013.

  • Description: We will study explicit results and theorems related to the density of nontrivial zeros on the critical line of the Riemann zeta function, one of the most important functions in analytic number theory.
  • Prerequisites: MAA 4212 and MAA 4402

10. Hyperbolic Geometry by James W. Anderson (Chapters 1–3)

  • Description: For about 2000 years, mathematicians could not settle the following question: does Euclid’s parallel postulate follow from the first four axioms of Euclidean geometry? In the early 19th century, Bolyai, Lobachevsky, and Gauss independently discovered hyperbolic geometry: a system of geometry in which the first four axioms of Euclid hold, while the parallel postulate does not. Hyperbolic geometry remains relevant in modern low-dimensional topology, complex analysis, and group theory. However, little background is needed to explore this alternate universe.This course intends to develop the properties of hyperbolic geometry while assuming minimal prerequisite knowledge. With a focus on the hyperbolic plane, we will construct the space and extensively discuss Möbius transformations. These transformations will motivate us to define the notion of arc length, distance, and isometries of hyperbolic space. Depending on the pace of the course, we may also discuss the Gauss-Bonnet formula and hyperbolic trigonometry.
  • Prerequisites: MAS4105 Linear Algebra, MAA4211/MAA4212 Real Analysis and Advanced Calculus I and II
  • Recommended background: Familiarity with group theory and the complex plane.

11. The Probabilistic Method by Noga Alon

  • Description: This is the book on the Probabilistic method one of the most important tools in combinatorics.
  • Prerequisites: Have seen proofs before. Recommended but not strictly necessary.

12. Sze-bi Hsu, Xiaoqiang Zhao, A Lotka–Volterra competition model with seasonal succession, J. Math. Biol. 64, 109–130 (2012). 

  • Description: Dynamic systems are an important topic used to describe a variety of time-evolving phenomena, most of which are purely continuous or purely discrete. Theoretical analysis of dynamical systems of periodic evolution is not that much, but it is an important topic. Biology and ecology often provide good models, and the model in this article is derived from the laws of seasonal evolution. This allows us to pay attention to the diversity of evolutionary phenomena, and also to use mathematical theory to analyze the long-term behavior of the system when the evolutionary law takes on a periodic form. This work is very helpful for undergraduate mathematics students, especially to help students establish their confidence in applying mathematical theories to practical phenomena. On the other hand, this work is also helpful for non-mathematics students, allowing them to understand the contribution of mathematics to real life, serving their future work, and establishing potential connections.
  • Prerequisites: MAP2302 and (MAS3114 or MAS4105)  are required. MAS4105, MAP4305 and MAP4314 are preferred.

 

For reading ideas, you may find it helpful to view readings from previous semesters.

Questions?

Email Konstantina Christodoulopoulou (kchristod@ufl.edu) or Sara Pollock (s.pollock@ufl.edu).