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.