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Pictured: The 2024 Directed Reading Mentors and Mentees
Welcome to the directed reading at du math webpage!
Welcome to the directed reading at du math webpage!
This year's edition of the Directed Reading at DU program will run through the
Winter and Spring Quarters 2026.
If you would like to get an idea of what happened in earlier editions, you can find this from the menu!
The goal of the program is twofold:
to provide opportunities and support for undergraduates interested in learning beautiful math and math-adjacent topics under the guidance of one of our graduate student mentors;
to provide our graduate students the opportunity for mentoring undergraduate projects.
We will focus on creating a welcoming learning community for the exchange of beautiful mathematics. We will also work on presentation skills with the goal of communicating math ideas in an effective way.
New for the 2026 edition: Mentors and Mentees will receive small stipends!
The stipends are made possible by a CAREER grant through the U.S. National Science Foundation (DMS-2439897).
MEET THE MENTORING TEAM FOR THE 2026 EDITION OF THE DIRECTED READING PROGRAM!
Luke Hetzel
Eden Ketchum
Christian Naess
Paul Johnson
Kaya Wright
Zion Hefty
Mandi Schaeffer Fry - faculty mentor
The 2026 DR Projects:
The 2026 DR Projects:
Word-Representable Graphs - Mentor Zion Hefty
Word-Representable Graphs - Mentor Zion Hefty
(Prerequisites: Formally, none; however, some familiarity with proofs would be helpful)
A graph is a collection of vertices and edges connecting some of those vertices. A word is a finite string of letters. We will study word-representable graphs, graphs for which there exists a certain kind of word associated with them (more precisely, they are graphs for which there exists a word on the vertex set of the graph such that an edge xy is present in the graph exactly when x and y alternate in the word). Along the way, we will learn about partially ordered sets and orientations on graphs, and how those are useful tools when determining if a graph can be represented by a word in this way, and if so, how long/short such a word must/can be.
Book: Words and Graphs by Sergey Kitaev and Vadim Lozin
Introduction to Number Theory - Mentor Eden Ketchum
Introduction to Number Theory - Mentor Eden Ketchum
(Prerequisites: Formally, none; however, some familiarity with proofs could be helpful)
The topic of number theory has been of interest to mathematicians for thousands of years. In recent times it has proven to be useful for computer science and cryptography. In this reading cover the following topics associated with the set of integers: divisibility, congruences, modular arithmetic, arithmetic functions and further topics based on the interest of the student.
Book: Elementary Number Theory by Burton
Hopf Algebras and String Diagrams - Mentor Paul Johnson
Hopf Algebras and String Diagrams - Mentor Paul Johnson
(Prerequesites: linear algebra; some abstract algebra (such as knowing what a group is) is highly recommended.)
Sitting behind familiar algebraic structures like groups and rings lies structural properties such as commutativity and associativity. These properties can be represented visually with the language of string diagrams, as the fusing and twisting of strands. Hopf algebras generalize a lot of algebraic objects, in a way that can be twisted and deformed, creating a bridge to knot theory. String diagrams themselves are also useful, for example they provide a visual way to think about quantum computing. We will be following the book "A Course on Hopf Algebras" by Rinat Kashaev, although we can divert to talk about other topics.
Nonstandard Analysis and the Infinitesimal Calculus - Mentor Christian Naess
Nonstandard Analysis and the Infinitesimal Calculus - Mentor Christian Naess
(Prerequisites: just calculus, but real analysis would be a bonus!)
The study of the differential and integral calculus was initiated in the seventeenth and eighteenth centuries, mainly by Isaac Newton and Gottfried Leibniz. This early treatment of the subject relied on an intuitive notion of an "infinitesimal" a number smaller than any positive real number but importantly non-zero.
As such a number does not exist within the real numbers, when calculus was formalized in the nineteenth century, as an analysis of the real number line and its functions, the intuitive infinitesimals were rejected in favor of the so called ε,δ approach. It was not until the 1960's that Abraham Robinson formalized a sort of "nonstandard analysis" that uses a rigorous notion of infinitesimal, a type of hyperreal number, in order to give a rigorous foundation to the intuitive constructions of Newton and Leibniz.
The benefits of the infinitesimal calculus aside from justifying the historical origins of the subject are mainly pedagogical and technical. On the side of pedagogy, some mathematicians and educators argue that the infinitesimal approach is more easily grasped by students than the usual ε,δ approach. On the technical side the method has had use in pure and applied mathematics, including physics and economics as a source of mathematical models.
Book: Foundations of Infinitesimal Calculus by H. Jerome Keisler.
Introduction to Cryptogrophy - Mentor Luke Hetzel
Introduction to Cryptogrophy - Mentor Luke Hetzel
(prerequisites: formally, none; but linear algebra and/or some coding experience could help!)
From spies and generals to banks and businesses, encryption is essential. Much of the Allies' success in the second world war can be attributed to their breaking of the enigma machine. The reason only you can access your bank account is the strength of RSA protocols. In this directed reading we will study various methods of encryption and decryption that have been used throughout history. Depending on student interest we may study modern computer assisted methods of encryption and decryption as well.
Algebraic Combinatorics - Mentor Kaya Wright
Algebraic Combinatorics - Mentor Kaya Wright
(prerequisites: linear algebra; some group theory could also be helpful but is not a strict requirement)
Did you enjoy linear algebra? Do you wish to hear of matrices and eigenvalues and graphs and counting with an algebraic flavor throughout?
Come! Join me on a read through of Algebraic Combinatorics: Walks, Trees, Tableaux, and More by Richard P. Stanley
We'll start off with a discussion of counting walks in graphs using matrices and eigenvalues and then move on to other topics connected to graphs and partitions as time allows.
More details about the program:
During the Winter and Spring quarters, mentees will meet once a week for an hour with their mentors to discuss material related to their project.
At the end of the program (near the end of the spring quarter), each mentee will give a 20-minute presentation on something cool that they have learned. The mentor-mentee team will work together on this presentation during the last month of the program.
We will have a few social events in which we all get together to give updates on our projects and to consider relevant issues to the program such as: stories of resilience in mathematics, ways in which we can build a welcoming math community, ways in which we can improve as communicators of mathematical ideas, and ways in which we can be supportive mentors.
If you are on the fence about applying and/or have any questions please reach out to Dr. Mandi (mandi (dot) schaefferfry (at) du (dot) edu).
Prerequisites to apply:
1) Be excited about at least one of the projects described above. (Please read the project descriptions to make sure that you satisfy any class specific prerequisites of the project. Please, also note that most projects have no extra prerequisites).
2) Be available to meet once a week for an hour with your mentor, plus 1.5 hours of reading/work outside of the weekly meeting during the Winter and Spring quarters.
3) Be available for a few social gatherings.
4) Be interested in presenting some of what you have learned at the end of the program with the help and support of the mentoring team.
5) Be an undergraduate at DU.
APPLICATIONS ARE OPEN FROM October 17th TO NOV 7TH, 2025
Read the project descriptions above and
APPLY HERE!
We will be in touch with news on the mentor-mentee pairings in November 2025.
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