Department of Mathematics
University of Georgia

## Summer REU: Knots and Algebra

Leaders: Akram Alishahi and Melissa Zhang

Dates: June 1, 2021 -- July 9, 2021

Description: A knot is a tangled-up piece of string with the ends glued together. A useful way to depict a knot is to draw a knot diagram, which is a projection of the knot onto the plane, as if one had snapped a picture of it with a camera. While knot diagrams are very useful to look at, they bring up some interesting questions: How can we tell when two knots are actually the "same", i.e. we can wiggle one of them to look exactly like the other? For that matter, how can we tell when a knot is actually unknotted, i.e. just a regular circle without any kinks in it? How many times must a knot "pass through itself" for it to become unknotted? Even the simplest questions about knots can sometimes be surprisingly difficult to answer. Nevertheless, in the past century, we have developed a variety of different types of tools for studying knots, ranging from studying topological surfaces with knotted borders to constructing algebraic representations of knots. Moreover, much can be learned by simply drawing pictures of knots and mulling over them.

The aim of this project is to introduce students to some of the basic tools and constructions in the field of knot theory with the goal of generating new ideas for approaching some surprisingly open questions. Along the way, students will learn how to interface with existing literature and how to communicate ideas to the scientific community.

Activities: Students will spend an intensive 6 weeks engaged in collaborative activities such as the following:

• Lectures and discussions: Students will learn to use basic tools in knot theory.
• Problem sessions: Students will put their new skills to practice by working together to solve problems with known solutions.
• Research reports and presentations: Students will discuss their progress and improve their math communication skills.

Background / Prerequisites: Students are expected to have some experience with linear algebra, and to have an interest in studying unsolved problems in mathematics. While a formal course in proof-writing is not strictly required, the well-prepared participant should be willing to think carefully and deeply, and to be able to express their ideas clearly.

Due to the uncertainties of the current pandemic, this is a virtual program, so students should make sure they have a reliable internet connection during the weeks of active research.

We are committed to providing a positive and welcoming environment for all students, especially those from groups that have been historically underrepresented in academic mathematics. All undergraduate students are welcome to apply.

Stipend: Students will receive a \$4000 stipend.

Application: Interested students should email the following to akram.alishahi@uga.edu and melissa.zhang@uga.edu. Make sure the subject of your email includes the phrase "Summer REU Application" to ensure that your application is received. We will notify you upon receipt of your application.

• Unofficial transcript
• A personal statement (no more than one page):
• Tell us about yourself: What topics and subjects are you interested in? What courses have you most enjoyed?
• Tell us why you are interested in participating in this REU, and how this experience would help you achieve your future goals.
• If you have any relevant research or work experience, or relevant skills, feel free to let us know.
• Finally, we expect our students to work together as a group. Briefly explain how you have worked well with others in the past, and how you would help the group work together smoothly.
• Name(s) of one or two professors the student has had contact with to serve as reference