So far, I have spent much time thinking about 4-dimensional hyperkähler ALE spaces. To be precise, I gave a new gauge-theoretic construction of these spaces by realizing them as moduli spaces of solutions to certain PDEs, partially motivated by Peter Kronheimer's original construction. Currently, I'm trying to generalize the gauge-theoretic construction to get a larger class of moduli spaces. I hope to study the topology of these moduli spaces using tools from gauge theory and be able to see the interactions between low-dimensional topology, representation theory, and singularity theory through this perspective.
I am also interested in learning about minimal surfaces, instanton Floer theory, Heegaard Floer theory, Seiberg-Witten theory, symplectic and algebraic geometry, and homotopy theory.
Papers
Yan, J. A new gauge-theoretic construction of the 4-dimensional hyperkähler ALE spaces. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02944-3. PDF
Conference and Seminar Talks
Braids Reunion Workshop, ICERM, July 15-19, 2024
AMS Southeastern Sectional Meeting, Florida State University, March 23-24, 2024
Geometry Seminar, University of Virginia, March 12, 2024
Geometry Seminar, University of Maryland College Park, March 4, 2024
Geometry Seminar, Rice University, January 17, 2024
The Interactive Workshop, Erdös Center/Rényi Institute, June 19, 2023
Summer School on Involutions, University of Nantes, May 30, 2023
Low-Dimensional Topology Seminar, MPIM, May 11, 2023
Gauge Theory Virtual, Virtual, January 10, 2023
Knots in Washington 49.875, Washington DC, December 9-11, 2022 (Distinguished Graduate Student Presenter)
MSRI Gauge Theory Graduate Student Seminar, Berkeley, October 18, 2022 (Recording)
Interactions between 3- and 4-Manifolds at AMS Southeastern Sectional Meeting, Chattanooga, October 15-16, 2022
ICERM Braids Graduate and Postdoc Seminar, Providence, May 2, 2022
AMS-AWM Special Session: Women and Gender Minorities in Symplectic and Contact Geometry at JMM, Virtual, April 6-9, 2022
*A more comprehensive list of talks that I have given can be found on my CV.
Co-organized Seminars and Workshops
ICERM Braids Graduate and Postdoc Seminar, Providence, February 1- May 6, 2022