# Research

*This page will no longer be updated starting from 01/16/2018. Jump to here for my most up-to-date information on research.*

13. *Symplectic rational G-surfaces and equivariant symplectic cones. ***Submitted, ***arXiv:1708.07500. *(with Weimin Chen, Tian-Jun Li)

We give characterizations of a finite group G acting symplectically on a rational surface (\CP^2 blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of G-conic bundles versus G-del Pezzo surfaces for the corresponding G-rational surfaces, analogous to a classical result in algebraic geometry.

Besides the characterizations of the group G (which is completely determined for the case of \CP^2#N\overline\CP^2, N=2,3,4), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given G-rational surface.

12. *Symplectic −2 spheres and the symplectomorphism group of small rational 4-manifolds. arXiv:1611.07436. *(with Jun Li, Tian-Jun Li)

We study both connected components and fundamental groups of the symplectomorphism groups for small rational surfaces. Especially, when the rational surface has Euler characteristic eight, we computed the symplectic mapping class groups for all symplectic forms, and related them to braid groups on a sphere.

11. *Dehn twists exact sequences through Lagrangian cobordism.** *** Submitted*** , arXiv:1509.08028. *(with Cheuk-Yu Mak)

We give a new point of view to various long exact sequences in the symplectic literature. In particular, we interpret them as a direct consequence of a clean surgery construction and Biran-Cornea's Lagrangian cobordism theory. This is a "mirror" construction of the Fourier-Mukai construction of spherical/projective twists in algebraic geometry. We also obtained a Floer-theoretic description of Lagrangian projective twists, confirming a mirror conjecture of Huybrechts and Thomas.

Besides these, we introduced a "bottleneck formulation" to immersed Lagrangian cobordisms, and a new simple algebraic technique which is very handy at computing connecting maps coming from cobordisms in certain cases.

10. *Gauged Floer homology and spectral invariants*. * ***IMRN**, to appear arXiv:1506.03349. (with Guangbo Xu)

We established the spectral theory in the gauged Floer setting. Entov-Polterovich's quasi-morphism and quasi-state theories were defined in this case. As applications, we obtained a weak Arnold conjecture in arbitrary toric manifolds without appealing to virtual techniques, and also Expanded the class of superheavy Lagrangians.

9. *Equivariant split generation and mirror symmetry of special isogenous tori. ***Advances in Mathematics*** , *to appear,

**arXiv:1501.06257.**We established the homological mirror symmetry for a class of symplectic tori called the "special isogenous tori". Also we showed by Orlov's criterion on the $B$-side, that the derived Fukaya category is a complete invariant in this class of symplectic tori. The main technical ingredient is a generalization of Abouzaid's generation criterion to an equivariant version when a free finite group action is present; some classical results of lattice quotients in rigid analytic geometry were also employed.

8. *Stability and existence of surfaces in symplectic 4-manifolds with b+=1*. **J. Reine Angew. Math. (Crelle's Journal), **to appear, arXiv:1407.1089. (with J. Dorfmeister and T.-J. Li.)

We completely classified the homology class of smooth and symplectic (-4)-spheres in rational and ruled surfaces, and established the existence of ADE-plumbing of Lagrangian spheres under minimal assumption. Two main technical innovations were made: we extended part of Opshtein-Mcduff's non-generic Gromov-Witten technique to more complicated non-generic configurations, and we introduced a "tilted transport" to construct symplectic submanifolds.

7. * Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians*. **J. Symp. Geom.**, to appear, arXiv 1305.7291. (with R. Hind and M. Pinsonnault)

We established the long unproved result that the space of Lagrangian $\RP^2$ in $T^*\RP^2$ is contractible. The same method should also yield an easy proof for the $S^2$ case due to Richard Hind. Relations to the homotopy type of symplectomorphism groups of certain non-compact symplectic manifolds with concave ends, as well as space of orbifold embeddings.

6. *The symplectic mapping class group of $\CP^2\#n\overline{\CP^2}$, \leq 4$*. **Michigan Math. J.**, 64 (2015), 319-333. Journal link. arXiv:1310.7329. (with J. Li and T.-J. Li)

We established the connectedness for symplectomorphism groups for small rational surfaces.

5. * On an exotic Lagrangian torus in $\CP^2$*. **Compositio Math. ** 151 (2015), 1372-1394, Journal link. arXiv 1201.2446.

This paper studies a semi-toric system on $\CP^2$ with an $\RP^2$-singularity. We found a superheavy fiber in the sense of Entov-Polterovich using the Fukaya-Oh-Ohta-Ono's framework, which provides a negative answer to their question asking whether $\RP^2$ is a stem in $CP^2$. The main innovation is to develop a new approach to count holomorphic disks with Lagrangian fiber boundaries in such a toric system with singularities.

4. *Exact Lagrangians in A_n-singularities*. **Math. Ann.**, 359 (2014), no. 1-2, 153-168. Journal link, arXiv 1302.1598.

We prove two longstanding open problems: the isotopy classification of compactly supported symplectomorphisms and that of Lagrangian spheres in A_n-surface Milnor fiber. The main technique we used is the construction of a ball-swapping symplectomorphism, as a symplectic incarnation of classical monodromy maps in algebraic geometry.

3. * Spherical Lagrangians via ball packings and symplectic cutting*. **Selecta Math. (N.S.)**, 20 (2014), no. 1, 261-283. Journal Link, arXiv 1211.5952. (with M. S. Borman and T.-J. Li)

Several long-term open questions on uniqueness of Lagrangians submanifolds were concluded in this paper: this includes smooth isotopy and symplectomorphic uniqueness of Lagrangian spheres and real projective plane embeddings into rational or ruled manifolds in dimension 4. We also established a smooth knotted example for real $\RP^2$-embedding when the symplectic form is allowed to vary.

2. * Lagrangian spheres, symplectic surfaces and symplectic mapping class groups*. **Geom. & Topo. **16 (2012), 1121-1169. Journal Link, arXiv 1012.4146. (with T.-J. Li)

This paper works on Lagrangian spheres in symplectic 4-manifolds. Results in several different directions were obtained:

▸ We classified completely the homology class of Lagrangian $S^2$ embeddings into rational or ruled surfaces.

▸ We factorized the homological action of any symplectomorphism of rational or ruled surfaces into composition of Lagrangian Dehn twists.

▸ New short proof to Lagrangian embeddings of $S^2$ into $S^2\times S^2$ and $\RP^2$ into $\CP^2$ and $T^*\RP^2$.

▸ Hamiltonian uniqueness of $S^2$ in rational surfaces with Euler number less than 8.

1. * Note on a theorem of Bangert. ***Acta Math. Sin. (Engl. Ser.)** 28 (2012), no. 1, 121-132. Journal Link, arXiv:1509.08128. (with T.-J. Li)

This note is a preliminary attempt to establish uniruledness for a complete non-compact symplectic manifold, equipped with a **non-cylindrical** almost complex structure. We establish the non-hyperbolicity property of almost complex structures on an asymptotically standard symplectic manifold.

**Work In Progress:**

- Dehn twists along Lagrangian spherical space forms. (with C.Y. Mak)
- Finite symplectic symmetries and orbifold cohomology. (with L. Amorim, S.C. Lau)
- Donaldson hypersurfaces in orbifold theory. (with G. Xu)