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 delve into 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 – Fall 2025
- Graduate Mentor Applications closed: August 11.
- Undergraduate Mentee Applications close: August 26, 2025 at 2pm.
- Pairs/Groups Announced: August 27, 2025.
- Reading Period: August 27-December 3.
- Final Presentations: TBA
How to Apply
Undergraduates: Undergraduate Application Form
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: The Graduate Mentor Application is closed
Fall 25 Reading Projects
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- Increasing paths in countable graphs, by Andrii Arman, Bradley Elliott, and Vojtěch Rödl.
- Description: The above paper is very undergraduate-friendly; a student who has taken an introductory course in combinatorics or graph theory should be able to understand it. The paper may also serve as an introduction to the study of infinite graphs and hypergraphs. Additionally, it contains a few open problems that are suitable for further investigation.
- Prerequisites: MAD 4203 (Introduction to Combinatorics 1) or MAD 4301 (Graph Theory)
- Random Matrix Theory and Its Innovative Applications, by Alan Edelman and Yuyang Wang; and a supplement text book, An Introduction to Random Matrices, by Alice Guionnet, Greg W. Anderson, and Ofer Zeitouni
- Description: My research interests lie in the study of bounded linear maps on infinite-dimensional Hilbert spaces (operator theory and free probability). A major breakthrough in the 1980s was the discovery of a connection between free probability and random matrix theory—in particular, the insight that large finite-dimensional random matrices can approximate the behavior of deterministic operators on infinite-dimensional Hilbert spaces. The goal of this reading course is to better understand the random matrix theory (RMT) aspects of this connection.The paper offers an accessible and computationally grounded survey of RMT, with an emphasis on modern applications and numerical experimentation. The authors guide the reader through key RMT phenomena such as eigenvalue repulsion, universality, the Marchenko–Pastur law, and the semicircle law. My hope is to engage with the computational aspects of the paper as well, by running the numerical experiments to visualize and explore these phenomena firsthand.
- Prerequisites: MAS4015 and some computer programming experience
- Recommended Background: Interest in Probability
- Notes on Schubert polynomials, by Ian G. MacDonald
- Description: This will be a directed reading in algebraic combinatorics. We will use combinatorial models for polynomials to try to solve some expansion problems in Schubert calculus.
- Prerequisites: Graduate algebra or combinatorics.
- Recommended Background: Familiarity with permutations.
- An Introduction to Manifolds, by Loring W. Tu
- Description: This reading group will explore Euclidean spaces and differential forms on R^n, following Chapter 1 of An Introduction to Manifolds by Loring W. Tu. Topics will include smooth functions on R^n, tangent vectors, vector fields, multicovectors, wedge products, differential forms, and the exterior derivative, with an emphasis on understanding these objects in a coordinate-free, algebraic framework. The goal is to build intuition and fluency with the tools of differential geometry in flat space, laying the groundwork for future study of manifolds; no prior exposure to differential geometry is assumed.
- Prerequisites: MAS 4105 and MAS 4301.
- Recommended Background: Any experience with real analysis or topology will be helpful.
- Primary: Topology and Data, by Carlsson, G. (Bulletin of the American Mathematical Society, 2009). Supplementary: Selected sections from Computational Topology: An Introduction, by Edelsbrunner, H. & Harer, J. (Chapters 7-8 on persistent homology)
- Description: Topological Data Analysis (TDA) is an emerging field that employs tools from topology to analyze the “shape” of data. Carlsson’s influential survey paper provides a foundation in fundamental concepts of TDA. This paper contains concrete examples and computational methods to enhance understanding. The reading material can also be paired with an applied project on real datasets for hands-on learning.
- Prerequisites: Linear Algebra (MAS 4105 or equivalent) and Abstract Algebra I (MAS 4301)
- Differential Geometry of Curves and Surfaces, by Manfredo P. do Carmo
- Description: I plan to cover the basic theory of curves and surfaces in R^3. This ideally would include the following sections of the above book: Chapter 1 sections 1-5, chapter 2 sections 1-6, and chapter 3 sections 1 and 2. In terms of content, this would cover the following topics: parametrized curves, arc length, the frame equations of Frenet and Serret, regular surfaces, differentiable functions and their differential, the first fundamental form, the Gauss map and its differential. Topics and emphasis may change according to interest. I intend to highlight the general ideas of the subject while encouraging students to make explicit calculations in coordinates.This material presents basic ideas of geometry before and during the time of Gauss. The material itself is highly useful in physics and engineering. However, the ideas of Gauss also suggested the possibility of an abstract formulation of geometry. This project was carried out by Gauss’ student Bernhard Riemann in the creation of modern differential geometry, a field of inestimable importance in science and mathematics. Surface theory as laid out by do Carmo gives one a strong geometric intuition and motivates the seemingly abstruse aspects of modern differential geometry (e.g. coordinates, tangent vectors presented as linear functionals, notions of curvature). Thus, a course in curve and surface theory gives a student valuable mathematical skills, while providing an excellent foundation to explore differential geometry further.
- Prerequisites: MAC2313; MAS4105 is highly recommended, but not strictly necessary.
- Recommended Background: Basic general topology and advanced calculus.
- Non-cooperative games, by J. Nash
- Description: The above as well as complementary material from the bibliography presented in the article will provide students with a wealth of resources to deeply understand and apply game theoretic concepts in mathematics and technology.
- Prerequisites: The students are expected to have some basic background in proofs and real analysis.
- Recommended Background: Basic Probability and Analysis
- Chapter 4, Navigational Complexity of Configuration Spaces, in the book “Invitation to Topological Robotics”, by Michael Farber.
- Description: This material has been chosen to introduce folks to the topic of topological robotics, which is at the center of mathematics and engineering. The crux of this new, actively-emerging field is to study topological invariants of topological spaces which are motivated and inspired by applications in robotics and engineering. The simplest example (which is discussed in the material) is the motion planning problem, which is a central theme in robotics. The problem is to construct a “nice” algorithm for a given robot that enables it to autonomously navigate between any input of positions in its configuration space. How difficult is it construct such an algorithm? The answer depends on the topology (or the shape and structure) of the configuration space of the robot, and the invariant, topological complexity (studied in the material), gives a numerical measures of that difficulty. In other words, it measures the complexity of the problem of navigation in a topological space, which is viewed as a configuration space of a robotic system. This material will help folks understand how this complexity is estimated and determined, how the problems related to this invariant are linked with other fascinating parts of mathematics, and how computing this invariant actually helps scientists and engineers in advanced motion planning (such as in planning collision-free motions of robots).
- Prerequisites: Elements of Set Theory, Introduction to Real Analysis (or equivalent), and Abstract/Linear Algebra
- Recommended Background: Elements of Topology (optional)
- Basic Coding for Finite Element Methods, Xiaoming He lectures.
- 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: Introduction to numerical analysis, numerical linear algebra.
- Recommended Background: might be good if using python or Matlab
- 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.
- 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)
- Winning Ways for Your Mathematical Plays by Elwyn Berlekamp, John Horton Conway, and Richard K. Guy
- Description: One of the most influential and important books in combinatorial game theory.
- Prerequisites: None
- Recommended Background: Likes games
- An introduction to Mathematical Epidemiology, by Maia Martcheva, Ch. 1,2,6; 2. Modeling and mathematical analysis of the Dynamics of HPV in Cervical Epithelial Cells, by Cruz Vargas-De-León; 3. Control of HPV infection leading to cervical cancer, by M. Imran
- Description: The motivation for selecting these materials is to gain a clear understanding of how mathematics connects to real world phenomena, particularly in biology. Through these readings, we can explore how to translate complex systems into mathematical frameworks and make predictions that contribute to solving pressing real world issues.
- Prerequisites: Basic Calculus and Ordinary Differential Equations, Linear Algebra
- Recommended Background: A willingness to read Numerical Optimization concepts
- Chapters 1-8 from Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, by Strogatz, S. H. (2024), Chapman and Hall/CRC.
- Description: Almost all phenomena in biology, chemistry, or physics are complex nonlinear systems that change over time. Understanding and being able to explain such phenomena not only satisfies intellectual curiosity but also greatly contributes to making a better world.
This reading proposal is designed to equip the reader with some basic mathematical knowledge to understand and explain specific wonders of the world that are ultimately nonlinear. The concepts to be learned include: flows, stability, fixed points, bifurcations, linearization, conservative systems, closed orbits, limit cycles, Poincaré maps, chaos, fractals, and strange attractors. These concepts provide a strong foundation for a typical mathematical modeler. - Prerequisites: Calculus 1-2, MAP2302 (differential equations), and linear algebra ( Jacobian matrix, eigenvalues and eigenvectors, and the divergence theorem).
- Description: Almost all phenomena in biology, chemistry, or physics are complex nonlinear systems that change over time. Understanding and being able to explain such phenomena not only satisfies intellectual curiosity but also greatly contributes to making a better world.
- The Heine Transformation. In: Ramanujan’s Lost Notebook, by Berndt, B.C., Andrews, G.E. (2009), Springer, New York, NY., Heine’s method and to transformation formulas, by Bhatnagar, G., Ramanujan J 48, 191–215 (2019).
- Description: This project aims to focus on learning the Heine’s method for obtaining a transformation formula between two basic hypergeometric series, closely following Andrews and Berndt (2009), and if time permits, some of the recent developments. It would give the student a glimpse into the beautiful theory of partitions, enumerative combinatorics and q-series.
- Prerequisites: Calc1 and Calc2
- Recommended Background: Some familiarity with infinite series and summations may be helpful, although not mandatory.
- p-adic Numbers: An Introduction, by Fernando Q. Gouvêa (Chapters 1-4) and Keith Conrad’s expository articles found on his website
- Description: I believe Gouvêa’s textbook is the most accessible introduction to p-adic numbers for undergraduates, as it takes the approach of generalizing the familiar absolute value on the real numbers to the p-adic setting. This perspective leads naturally to defining the p-adic absolute value on the rationals and completing them to obtain the field of p-adic numbers. The book is also filled with numerous exercises to check the reader’s understanding throughout, as well as SAGE code examples that facilitate concrete computations. For additional background or clarification on specific topics, we can draw on Keith Conrad’s expository notes, which provide concise, accessible explanations to fill in any gaps.
- Prerequisites: MAA 4211 and MAS 4301
- Recommended Background: MAS 4203 and MAS 4302 would be helpful.
- Increasing paths in countable graphs, by Andrii Arman, Bradley Elliott, and Vojtěch Rödl.
Questions?
Email Konstantina Christodoulopoulou (kchristod@ufl.edu) or Sara Pollock (s.pollock@ufl.edu).