# Publications

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*. * ***Submitted, **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. ***Submitted****, ** 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 $\CP^2\#n\overline{\CP^2}$, $n\leq 4$*. **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 $\CC P^2$*. **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:**

- (with W.-M. Chen and T.-J. Li) Symplectic Cremona maps.
- (with J. Li and T.-J. Li)
*Topology of symplectic rational surfaces and -2 spheres*