2025 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 quantum topology, a subject at the juncture of low-dimensional topology and quantum field theory/string theory. In particular, the REU participants studied PGL(2) skein algebras of surfaces, a relatively unexplored but nonetheless important class of skein algebras. This was done 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

Papers

• PGL(2) skein algebras of small surfaces, Cole Ellison, Vijay Higgins, Mark Sandey, Andrew Sylvester.
• PGL(2) webs at roots of unity, Vijay Higgins, Bryan Silva, David Stephens, Amy Somers, Srijan Sundar, Parth Wokhlu.
• Bases for PGL(2) skein algebras and a presentation for the 4-punctured sphere, Cole Ellison, Vijay Higgins, Mark Sandey, Andrew Sylvester.

Talks

• JMM, Washington D.C., Cole Ellison, Mark Sandey, Bryan Silva, Amy Somers, David Stephens, Andrew Sylvester.
• Undegraduate Mathematics Symposium at UIC, Amy Somers, Parth Wokhlu.
• UCSB Topology Seminar, Cole Ellison, Amy Somers, Andrew Sylvester.

Date and Location

The program dates were 4 August through 12 September. 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 (10 AM - 5 PM) to fully participate. Some weekdays used a remote format with 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. Because the students were supposed to be local, they were expected to commute to campus.

Mentors

The REU mentors were Vijay Higgins (RTG Postdoctoral Scholar, UCLA), Roman Krutowski (PhD student, UCLA), Jung Joo Suh (PhD student, UCLA), and Yan Tao (PhD student, UCLA). The program was overseen by Ko Honda (Professor, UCLA).

Schedule

The schedule was as follows:

• Week 1-2: Background on quantum topology of knots and links.
• Week 3-4: Start working on project(s).
• Week 5: Start wrapping up project(s).
• Week 6: Present the results in both a written report and an oral presentation.