Topic: In 2023, the research topic will be Knot Theory. This program provides 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 will showcase the results of their research through a written report and an oral presentation. During the project, students may also be asked to write software. The faculty mentors will clearly outline the expected outcomes at the beginning of the summer and work closely with students to ensure the successful completion of the final project. In addition to hands-on research, the REU will feature informative sessions run by faculty members on topics such as research methodology, career prospects in mathematics, and preparing for graduate school.

Applicants:

• The students must be US citizens or US permanent residents (green card holders). The program is being funded by an NSF grant and this is one of the grant requirements.
• The REU is designed to be local in nature, so the students must be living in the greater Los Angeles region and be able to commute to UCLA campus.
• The REU is only open to undergraduate students, so students should not have already received their Bachelors degree at the time of the REU.

Date and Location: In 2023, the program dates are June 26th to August 4th. The expected workload is 40hr/week for 6 weeks. The REU program will be conducted in a hybrid format, requiring participants to be physically present in Los Angeles and available during business hours (9 AM - 4 PM) to fully participate. On Mondays, Wednesdays, and Fridays, participants will meet in-person on the UCLA campus. On Tuesdays and Thursdays, they will meet remotely via Zoom.

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

Mentors: In 2023, the REU mentors are Hyunki Min (Hedrick Assistant Adjunct Professor at UCLA) and Konstantinos Varvarezos (Assistant Adjunct Professor at UCLA); the program will be overseen by Sucharit Sarkar (Professor at UCLA).

Schedule: The tentative schedule will be 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