directed reading
at du math
Part of the mentoring team from 2023 (year 1 of the program) in the math common room in the picture above. From left to right: Isis, Flor, Vlad and Jesse. (Credit: Casey Schlortt )
Welcome to the directed reading at du math webpage!
For info about the 2024 presentations, go here!
The second edition of the Directed Reading at DU program will run through the Winter and Spring Quarters 2024. If you would like to get an idea of what happened in the first edition (which ran through Winter and Spring Quarters 2023), go here!
Thanks to the support of the DU Math Department, each year, each mentor-mentee team member receives a personal copy of the team's book.
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 an inclusive and diverse 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.
MEET THE MENTORING TEAM FOR THE 2024 EDITION OF THE DIRECTED READING PROGRAM!
Casey Schlortt (she/her/hers)
Kempton Albee
Luke Hetzel
(He/Him)
Eden Ketchum
Di Qin
Paul Johnson
Mandi Schaeffer Fry - faculty mentor
(she/they)
The 2024 DR Projects:
Sturmian Subshifts - Mentor Casey Schlortt
(Prerequesits: Formally, none; however, some familiarity with proofs could be helpful)
Dynamical systems are mathematical models used to describe how a system (for example planetary movement or particle systems) evolves or changes . Symbolic dynamics is a branch of mathematics that deals with the representation and study of dynamical systems using sequences of symbols. Sturmian subshifts are an important class of symbolic dynamics, intriguing for both their intricate and simple properties. In this Directed Reading we will explore Sturmian subshifts and their importance in the field of symbolic dynamics.
Introduction to Dynamical Systems - Mentor Luke Hetzel
(prerequisites: none)
How do we describe the motion of the planets? At any single moment each planet has a mass, a position, a velocity and an acceleration. But then, due to the position of other planets and the laws of gravity, each planet’s position, velocity and acceleration will change. Then because of that change, how gravity acts upon this new “state” of the planets will change. And so on.
The theory of dynamical systems offers a tool for describing the behavior of complex evolving systems. It has applications in planetary motion, biological systems, particle motion, flow of water and even data storage.
This project will provide an introduction to the tools of dynamical systems. To do this we will first cover the basics of topology. We will then explore examples of dynamical systems which arise in various areas of mathematics.
Combinatorics - Mentor Di Qin
(Prerequesits: Formally, none; however, some familiarity with proofs could be helpful)
This directed reading will work through parts of the book Introductory Combinatorics by Richard Brualdi:https://www.pearson.com/en-us/subject-catalog/p/introductory-combinatorics-classic-version/P200000006138/9780137981045?tab=title-overview
The Algebra of Logic - Mentor Kempton Albee
(Prerequisites: Some familiarity with proofs)
In this directed reading, the student will learn about the connections of syntax and semantics in logic. In particular, they will study soundness and completeness results which will exploit the connection of formal proofs in a deductive system with equational truths in respective algebras. The main focus will be the propositional logic and it's algebraic counterpart Boolean algebras. However, depending on the pace and the understanding, I would like to tackle other classical logics such as modal and intuitionistic logics.
Elementary Number Theory - Mentor Eden Ketchum
(Prerequesits: Formally, none; however, some familiarity with proofs could be helpful)
This directed reading will follow the book Elementary Number Theory by Burton: https://link.springer.com/book/10.1007/978-1-4471-0613-5?source=shoppingads&locale=en-us&gclid=CjwKCAjw7oeqBhBwEiwALyHLM_0bTtYPrOclZignTYba-0H0MxyZ5wNp0QZeJxTLwSI4iscWrqt-vRoC5h8QAvD_BwE
Some possible final project topics could include: categorizing all Pythagorean triples or applying what the student has learned to basic cryptography.
Knots, Quandles, and Algebraic Structure - Mentor Paul Johnson
(Prerequesits: Linear Algebra encouraged, but a basic understanding of matrices and vectors will suffice. )
Knots, Quandles, and Algebraic Structure
Knots are some of the most universal objects in human art and aesthetics, and can be found in cultures throughout the world. They also show up everywhere in nature, from protein folding to quantum mechanics. What mathematics are hiding within knots, and how can we build a theory to characterize them? What does it mean to turn a geometric object into an algebraic structure?
Knot theory provides a way to formalize and characterize knots, leading to powerful invariants and beautiful classification theorems. Knot theory brings together concepts across many fields in mathematics, from abstract algebra to algebraic topology. In particular, a modern technique uses algebraic objects known as quandles, which have a self-distributive axiom in place of associativity.
This project will introduce the basics of knot theory and abstract algebra in a hands-on way. Knot properties will be shown using pictures and diagrams, and then algebraic structures will be developed through concrete examples on matrices. The goal will be a motivation of what an algebraic structure is and how to build one, rather than starting from axioms. With these tools, we will cover the basic theory of quandles and how they can be used with knots. Depending on how quickly we move, there could be time to talk about tools in algebraic topology applied to knots, at a very introductory level.
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, 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: ways in which we can build an inclusive and diverse math community, ways in which we can improve as communicators of math ideas, and ways in which we can be supportive mentors.
We are particularly interested in supporting the participation of students of color, including Latinx, Black and Indigenous students as well as the participation of LGBTQI+ students, non-binary students, trans students and/or women. 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 NOV 1ST TO NOV 15TH, 2023
Read the project descriptions above and
APPLY HERE!
We will be in touch with news on the mentor-mentee pairings in late 2023 or early January, 2024.