2023 REU in Geometry and Topology at UCLA

The REU was a six-week summer program offered by the Geometry and Topology group at UCLA and was funded by a NSF Research and Training Grant (RTG).

Topic

The research topic was Knot Theory. This program provided students with the opportunity to immerse themselves in active research projects in Knot Theory, an important branch of Low-Dimensional topology, under the supervision of senior and junior faculty in the field. During the final week of the program, students showcased the results of their research through a written report and an oral presentation. During the project, students also wrote software. The faculty mentors clearly outlined the expected outcomes at the beginning of the summer and worked closely with students to ensure the successful completion of the final project. In addition to hands-on research, the REU featured informative sessions run by faculty members on topics such as research methodology, career prospects in mathematics, and preparing for graduate school.

Outcome

• Positive 2-bridge knots and chirally cosmetic surgeries, Michael Huang, Zelong Li, Rahi Tanaz, Chengyi Zhang.

Date and Location

The program dates were 2023-06-26 to 2023-08-04. The workload was 40hr/week for 6 weeks. The REU program was conducted in a hybrid format, requiring participants to be physically present in Los Angeles and available during business hours (09:00 to 16:00) to fully participate. On Mondays, Wednesdays, and Fridays, participants met in-person on the UCLA campus. On Tuesdays and Thursdays, they met remotely via Zoom.

Funding

Each student was provided a $5000 stipend for 6 weeks of REU at the end of the program. No additional funding was available for lodging or transportation. Since the students were supposed to be local, they were expected to commute to campus.

Mentors

The REU mentors were Hyunki Min (Hedrick Assistant Adjunct Professor at UCLA), Sumeyra Sakalli (Visiting Assistant Professor at University of Arkansas), and Konstantinos Varvarezos (Assistant Adjunct Professor at UCLA); the program was overseen by Sucharit Sarkar (Professor at UCLA).

Schedule

The schedule was as follows:

• Week 1: Introduction to knot theory
• Week 2: Computing knot invariants (Alexander, Jones polynomial)
• Week 3: Learning about special families of knots (2-bridge)
• Week 4: Dehn surgery and the cosmetic surgery conjecture
• Week 5: Write a python program to verify the conjecture
• Week 6: Present the results in both a written report and an oral presentation