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. Using this construction, I have been able to offer a new dynamical framework for understanding the McKay correspondence.
For future work, I aim to construct more general gauge-theoretic moduli spaces and utilize them to investigate the interactions between low-dimensional topology, symplectic field theory, representation theory, and singularity theory from this perspective.
I am also interested in learning about minimal surfaces, instanton Floer theory, Heegaard Floer theory, Seiberg-Witten theory, algebraic geometry, and homotopy theory.
Papers
Yan, J. A new gauge-theoretic construction of the 4-dimensional hyperkähler ALE spaces. Math. Ann. 391, 1009–1044 (2025). https://doi.org/10.1007/s00208-024-02944-3. arXiv: http://arxiv.org/abs/2310.10. PDF
Conference and Seminar Talks
Geometry Seminar, University of Virginia, April 21, 2026
Geometry Seminar, Georgia Tech, April 20, 2026
Undergraduate Colloquium, Rice University, April 13, 2026
Geometry and Mathematical Physics Seminar, University of York, October 7, 2025
AMS Spring Western Sectional Meeting, California Polytechnic State University, May 3-4, 2025
AMS Special Session on Invariants in Geometric Topology: Low Dimensions and Beyond, Clemson University, March 8-9, 2025
Geometry Seminar, Duke University, December 2, 2024
Geometry Seminar, Rice University, September 10, 2024
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