Research

My research is in symplectic topology and geometric analysis. Symplectic topology studies symplectic manifolds, which are nonlinear generalization of the classic phase space in Hamiltonian dynamics. In symplectic topology, I have applied adiabatic degeneration and singular gluing techniques to Cauchy-Riemann type equations, and proved relations between different symplectic invariants including Floer cohomology and quantumn cohomology ([OZ1], [OZ2]). Using holomorphic jet bundle, I proved the embedding property of J-holomorphic curves in Calabi-Yau manifolds for generic J ([OZ3]). Then I applied these techniques to differential geometry. Combined with heat kernel expansion and eigenfunction estimates, I obtained canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces by perturbing the heat kernel embedding map, thus gave a new proof of Nash’s isometric embedding theorem, with better understood extrinsic geometry ([WZ]). I also found certain universal jet relations of the heat kernel embedding map ([Z]). Combined with Dirac operator, I found relations among holomorphic curves, Seiberg-Witten invariants, and thin instantons in G₂ manifolds ([LWZ]). Using Hörmander’s L² method, I obtained new criterions of the solvability of Dirac type equations by curvature integrals, with application to the transversality of moduli spaces of holomorphic curves and instantons ([JZ]).

  • [OZ1] Floer trajectories with immersed nodes and scale-dependent gluing, J. Symplectic Geometry, Vol.9, No.4, 483-636, 154pp, 2011, arXiv: 0711.4187, with Yong-Geun Oh
  • [OZ2] Thick-Thin decomposition of Floer trajectories and adiabatic gluing, 94pp, to appear in Acta Mathematica Sinica, English Series, 2023, arXiv: 1103.3525with Yong-Geun Oh
  • [OZ3] Embedding property of J-holomorphic curves in Calabi-Yau manifolds for generic J, Asian Journal of Math, Vol.13, No.3, 323-340, 2009, arXiv: 0805.3581, with Yong-Geun Oh
  • [WZ] Isometric embeddings via heat kernel, J. Diff. Geom., (99), 2015, 497-538, arXiv: 1305.5613, with Xiaowei Wang
  • [Z] High-jet relations of the heat kernel embedding map and applications, to appear in 2021-2022 MATRIX Mathematical Research Institute Annals, arXiv: 1308.0410
  • [LWZ] Thin instantons in G2-manifolds and Seiberg-Witten invariants, J. Diff. Geom., (95), 2013, 419-481, arXiv: 1107.1947, with Conan Leung and Xiaowei Wang
  • [JZ] Solvability of the Dirac Equation, Advances in Mathematics, (320C), 2017, 451-474, arXiv: 1407.6936, with Qingchun Ji