RESEARCH STATEMENT

JINGYU ZHAO

1. Introduction My research focuses on the study of symplectic and its relationships with mirror symmetry and Hamiltonian dynamics. In 1985, Gromov’s work [12] on pseudoholomorphic curves revolutionized the field of symplec- tic geometry. A can be thought as a map u:Σ → M from a Riemann ¯ surface Σ to a (M, ω) that satisfy the Cauchy-Riemann equation ∂J u = 0 with respect to some almost complex structure J. The counts of such curves in M are later referred as the Gromov-Witten invariants, which have been shown to be powerful symplectic invariants of (M, ω). Another driving force of recent developments in is coming from the mirror symmetry conjecture. It relates the symplectic geometry of (M, ω) to the complex geometry of its mirror (M ∨,J). It is predicted by physicists [8] and later proved by Givental, Liu-Lian-Yau [11, 21] that the genus zero Gromov-Witten invariants of (M, ω) match with period integrals on the mirror. As these period integrals are easier to calculate on the com- plex manifold, it has been an early success in deriving Gromov-Witten invariants from mirror 4 symmetry in the case of the quintic 3-fold in CP . Given a smooth function H : M → R, or a Hamiltonian, on a symplectic manifold (M, ω), one defines the Hamiltonian vector field XH by ω(·,XH ) = dH and denotes the time-1 flow of 1 XH by φH , which is a symplectic automorphism of (M, ω). One of the fundamental questions in Hamiltonian dynamics is to study how many closed 1-periodic orbits of this Hamiltonian 1 vector field XH can have, or equivalently how many fixed points φH can possess. On a closed (monotone) symplectic manifold (M, ω), Floer made a groundbreaking contribution [14] to this 1 question by proving the Arnold conjecture, which states that the number of fixed points of φH is bounded below by the minimal number of critical points of a smooth function H on M. The method that Floer introduced is to define a chain complex generated by the fixed points of 1 1 φH and the differential counts the solutions u: R × S → M to the perturbed Cauchy-Riemann equation ∂su + J(u)(∂tu − XH (u)) = 0 for some choice of almost complex structure J on M. 1 The resulting homology is called the of φH . In subsequent discussions, I will review my research in answering the following questions: • How to derive and understand enumerative invariants, such as the Gromov-Witten in- variants of symplectic manifolds, from mirror symmetry? (See Section 2 below.) • From a dynamical point of view, how do the ranks of the Floer homology groups grow n under iterations φH for n ∈ N? (See Section 3 on page 4.)

2. Enumerative invariants from mirror symmetry 2.1. Backgrounds. Despite the various mirror symmetry predictions that have been verified by physicists [8], Kontsevich in 1994 proposed a rigorous mathematical formulation of the mirror symmetry conjecture, called the homological mirror symmetry (HMS) conjecture. It states that for a pair of mirror Calabi-Yau manifolds X and X∨, there is a (derived) equivalence between the Fukaya category of the symplectic manifold X and the category of coherent sheaves on

1 Research Statement Jingyu Zhao the complex manifold X∨. One intriguing question to ask is that: does the homological mirror symmetry conjecture implies enumerative mirror symmetry in the sense of Givental, Liu-Lian- Yau? Using ideas of Barannikov [5], Katzarkov-Kontsevich-Pantev [23] from noncommutative geometry, the work of Ganatra-Perutz-Sheridan [10] showed that homological mirror symmetry recovers enumerative mirror symmetry for Calabi-Yau hypersurfaces in projective spaces, which 4 for instance encodes the genus zero Gromov-Witten invariants of the quintic 3-fold in CP in terms of the period integrals that solve Picard-Fuchs equations on its mirror. Besides the Calabi-Yau case, the (homological) mirror symmetry conjecture has been extended to Fano symplectic manifolds as in [2–4]. One natural question to study is that if there is still a passage from homological mirror symmetry to enumerative mirror symmetry in the Fano case. The mirror of a Fano symplectic manifold is given by a Landau-Ginzburg (LG) model, that is, a quasi-projective complex algebraic variety X∨ together with a holomorphic function ∨ W : X → C (usually with isolated singularities). For instance, when X is a toric Fano manifold, ∗ n the LG model W :(C ) → C is given by the Hori-Vafa potential [19]. Advantages of considering toric Fano manifolds are that their Fukaya categories can be understood by studying the A∞- structures defined by Fukaya-Oh-Ohta-Ono [16] on the of the Lagrangian fibre L of the toric moment map, and the HMS conjecture can be proved by Abouzaid-Fukaya-Oh-Ohta- ∗ ⊗k ∗ Ono [1]. Such an A∞-structure is given by mk : H (L)[1] → H (L)[1] so that ∞ X M ∗ ⊗k ∗ mˆ = mk : H (L)[1] → H (L)[1] satisfies mˆ ◦ mˆ = 0. k≥0 k=0 In Fukaya-Oh-Ohta-Ono’s construction, the LG potential W appears naturally as the curvature term of the A∞-algebra, which is a function defined on the formal deformation space (or weak Maurer-Cartan space) of this A∞-algebra. Its value W (b) is determined explicitly by X odd (1) mk(b, b, ··· , b) = W (b)eL for b ∈ H (L), k≥0 ∗ even where eL is the unit in H (L). For each ambient even cohomology class b in H (X), one b can further define a new A∞-algebra structure {mk}k≥0 via bulk deformations considered in [17]. There is an analogous Maurer-Cartan equation to equation (1), which defines the bulk-deformed b b potential W associated to the A∞-structure {mk}k≥0 for each b. After bulk deformation, the potential W b gives rise to a formal deformation of the Hori-Vafa potential W as b varies in a formal neighborhood of the origin in Heven(X). For potentials with isolated singularities, the theories of K. Saito [27, 28], M. Saito [29] and C. Sabbah [26] define a Frobenius structure or flat structure on the (miniversal) deformation space U of the mirror potential W . Similar to the Calabi-Yau case, where the flat coordinates on U can be obtained by period integrals of certain relative homolomorphic form, these period integrals are called oscillatory integrals for LG models. One interesting question to study is whether these oscillatory integrals have enumerative meanings in terms of Gromov-Witten invariants on the symplectic side.

2.2. My results. In a joint work with H. Hong and Y.-S. Lin, we answer the above question affirmatively when X is a toric Fano surface. Inspired by the previous work of M. Gross [13], 2 who matched tropical descendant disk counting with oscillatory integrals when X = CP , we computed the bulk-deformed potential W b for b in H4(X) associated to any Lagrangian torus fibre L of the SYZ fibration π. Given k generic point constraints p1, ··· , pk in X, the kth-order 2 2 potential Wk is defined as the counts of J-holomorphic disk u:(D , ∂D ) → (X,L) in some class β ∈ H2(X,L) passing through p1, ··· , pk and one generic point on L. The virtual dimension Research Statement Jingyu Zhao of this equals to dim(L) + µ(β) − 3 + 2k − 4k − 1. As when µ(β) = 2k + 2 the moduli space has virtual dimension zero, our bulk-deformed potential Wk can be thought as the counts of higher Maslov disks. Although the original Hori-Vafa potential is a well-defined Laurent polynomial, we discover that the bulk-deformed potential Wk experiences discontinuities between different chambers of the base of the SYZ fibration. Such phenomenon is called wall- crossing, which is caused by the existence of Maslov zero disks in Floer theory [2]. In particular, for non-generic Lagrangian torus fibre, there exists J-holomorphic disk with boundary on L passing through p1, ··· , pk that behaves as Maslov zero disks. We refer such disks as generalized Maslov zero disks. The locus in the SYZ base of Lagrangian torus fibers that bound such disks are called walls. To compute the bulk-deformed potential Wk, it suffices to determine the holomorphic wall structure.

2.3. Our techniques: A tropical-holomorphic correspondence Theorem. Our strategy to compute Wk is to use tropical geometry in which case the wall structures from (generalized) Maslov zero tropical disk counting can be determined combinatorially. To obtain the holomor- phic wall structure and compute Wk for all k ∈ N, we proved the following theorem. Theorem 2.1 (Hong-Lin-Z.). For a toric Fano surface X, the wall structure from generalized Maslov index 0 disks is homeomorphic to the one given by tropical disk counting. 2.4. A big quantum period theorem. Based on the recent work of D. Tonkonog [36], we also proved the following big quantum period theorem. Theorem 2.2 (Hong-Lin-Z.). Let X be a toric Fano surface and let L be any Lagrangian torus fibre of the moment map π. The bulk-deformed potential Wk associated to L satisfies 1 Z dx dx X X −m Wk/u 1 2 1 + hα, ··· , α, τm−2(α)iX,m+n,βu = 2 e ∧ (2πi) 2 x1 x2 0≤n≤k m≥2 T 4 for all k ∈ N, where α = PD([pi]) ∈ H (X) for i = 1, ··· , n + 1 and the formal variable u is usually referred as the descendant variable in Gromov-Witten theory. The above notation h·, ··· , ·iX,d denotes the Gromov-Witten correlation function Z X d1 ∗ dn ∗ hτd1 α1, ··· , τdn αniX,d = ψ1 ∪ ev1(α1) ∪ · · · ∪ ψn ∪ evn(αn). M0,n(X,β) β : c1(M)(β)=d

Theorem 2.2 gives a direct relationship between the oscillatory integrals of Wk and the de- scendant Gromov-Witten invariants of X. The later defines a cohomological field theory, or a formal Frobenius manifold structure on the symplectic side (c.f. [24, Chapter 3.4]).

2.5. The Gauss-Manin system on S1-equivariant Floer cohomology. In my earlier work 1 ∗ [39], I studied the S -equivariant Floer cohomology HFeq(M) for open symplectic manifolds (M, ω) (with convex boundaries), which was introduced by P. Seidel in [30] and defined explicit by Bourgeois and Oancea in [7]. When M is not exact, the S1-equivariant Floer cohomology ∗ P λi HFeq(M) is a module over ΛQ[[u]], where ΛK = { i aiq | ai ∈ K, λi → ∞ as i → ∞} is the Novikov field with ground field K. For certain class of admissible Hamiltonian Hk of slope ∗ k ∈ N, I proved in [38] that after inverting u to eliminate the u-torsion in HFeq(M), the PSS homomorphism [25] gives a canonical trivialization of the localization of S1-equivariant −1 ∗ Floer cohomology u HFeq(M,Hk) as a ΛQ((u))-module, which can be viewed as a localization Theorem for the circle action on the loop space. Research Statement Jingyu Zhao

Theorem 2.3 (Zhao [38]). Let M be a monotone (or exact) symplectic manifold (M, ω) with convex boundary, the PSS homomorphism induces an isomorphism after inverting u, that is, ∗ =∼ −1 ∗ PSS: H (M;ΛQ)((u)) −→ u HFeq(M,Hk), ∀k ∈ N. m ∗ Theorem 2.3 implies that after multiply some u for some m ≥ 0, any classes in s ∈ HFeq(M) ∗ can be written as a series in H (M;ΛQ)((u)), predicted by Theorem 2.2, this series should be determined by the genus zero Gromov-Witten invariants with descendants. 2.6. Proposed future research.

∗ 2.6.1. The connection ∇∂u on HFeq(M) and wall-crossings. Recently, P. Seidel [33] constructed the analogue of the Gauss-Manin connection ∇∂q and the connection in the u-direction ∇∂u ∗ ∗ on HFeq(M). It is natural to ask if we can find flat sections seq(q, u) in HFeq(M) satisfying −1 ∗ ∇∂q seq = ∇∂u seq = 0. The connection ∇∂u is defined on u HFeq(M) as c (M) ∗ (s) Gr(s) ∇ (s) = ∂ (s) − 1 + , ∂u u u2 2u where ∗ is the quantum product and Gr(s) = k · s if s ∈ Hk(M)((u)). Unlike the case of Calabi-Yau mirror pairs for which the connection ∇∂u is trivial, for Fano manifolds M this connection has irregular singularities (a second order pole) at u = 0. For the mirror LG model ∨ W : X → C, it is proved in [23, Lemma 3.11] that the asymptotic expansions of the oscilla- tory integrals experience the wall-crossing phenomenons as u → 0 in different angular sectors 1 ∗ Arg(u) ∈ S as one takes u valued in C . This is referred as Stokes phenomenons in [23][26]. I am interested in understanding this phenomenon explicitly in terms of the enumerative mean- 1 ∗ ings of the S -equivariant Borman-Sheridan classes seq in HFeq(M) considered in [32, Section 3c] on the symplectic side. 2.6.2. Enumerative meaning of Saito’s primitive form. Given an unfolding W˜ of the LG model ∨ W : X → C having isolated singularities, Saito’s theory of primitive forms states that there exists a family of volumn forms η(x , t ) = dx1 ∧ · · · ∧ dxn + O(t ), where x are coordinates on i i x1 xn i i ∨ X and ti are coordinates on a formal neighborhood of W in U, characterized by the following equation Z ˜ ∂ ∂ 1 k  W (xi,ti) ˜ + Aij(τ) e u η(xi, ti) ≡ 0, ∀ Γ ∈ Hn(X, Re(W /u)  0; C), ∂τi ∂τj u Γ n k where {τi(ti)}i=1 are flat coordinates on the universal folding space U and Aij(τ) are the struc- ture constants of the associative product on the tangent space of U. Inspired by the work of Iritani [20], one might wonder if the primitive form has an enumerative meaning in terms of Gromov-Witten invariants. Based on the proof of Theorem 2.2, I will investigate a conjectural relationship between the primitive form and (bulk-deformed) open Gromov-Witten invariants with descendants on a fixed Lagrangian brane L in toric manifolds.

n 3. Hamiltonian Dynamics: Prime iterations of the φp 3.1. Backgrounds. The realm of symplectic geometry also has great intersections with Hamil- tonian dynamics. Given a monotone (or exact) symplectic manifold (M, ω) with convex bound- ary and a compactly supported (exact) symplectomorphism φ: M → M, from a dynamical point of view, one might wonder how the dimensions of the Floer cohomology groups HF ∗(φn) grow under iterations of φ for n ∈ N. More preliminarily, even though one has the obvious relation that Research Statement Jingyu Zhao

N(φ) ≤ N(φ2) ≤ N(φ3) ≤ · · · , the corresponding relation for the rank of the Floer cohomology groups HF ∗(φn) is not obvious. 3.2. My results. Together with E. Shelukhin, we studied the behaviors of Floer cohomology ∗ pn groups for the prime iterations of a HF (φ ) for all n ∈ N . For any fixed prime p, there is a /p-equivariant Floer cohomology group HF ∗ (φp;Λ ) associated to the Z Z/p Fp Z/p-action on ∞ p σ Lφp (M) := {x(t) ∈ C (R,M) | x(t) = φ (x(t + p))}, (σx)(t) 7→ φ (x(t + σ)) for σ ∈ Z/p. For this Z/p-action on Lφ(M), we proved an analogue of the Smith inequality studied by P. Smith [35] in algebraic . In particular, it provides an rank inequality between Floer (co)homology groups of φ and φp. Theorem 3.1 (Shelukhin-Z.). Given a symplectomorphism φ ∈ Sympc(M) on a monotone or exact symplectic manifold with convex boundary and assume that fixed points of φpn are non- degenerate for all n ∈ N, there is a Smith inequality between dimensions of ungraded vector spaces over ΛFp ∗ p ∗ dim HF (φ ;ΛFp ) ≥ dim HF (φ;ΛFp ). The case of p = 2 has been proved in the exact case by K. Hendricks [18] and P. Seidel [31], our proof for general prime p > 2 is similar to the approach given by P. Seidel. c 3.3. Application on the symplectic mapping class groups. Let π0(Symp (M)) be the group of compactly supported symplectomorphisms up to Hamiltonian isotopies, or the sym- plectic mapping class group of M. One consequence of Theorem 3.1 is that if φ has a non-trivial ∗ ∗ pn fixed point, for instance if dim HF (φ) > dim H (M), so does its iteration φ for any n ∈ N. Therefore, all the iterations of φpn cannot be Hamiltonian isotopic to the identity. Theorem 3.2 (Shelukhin-Z.). Given a symplectomorphism φ ∈ Sympc(M) with non-degenerate fixed points, if dim HF ∗(φ;Λ ) > dim H∗(M; ), ΛFp Fp Q then all the pnth-iterations [φp], [φp2 ] ··· have non-trivial classes in the symplectic mapping class c group π0(Symp (M)) for any prime p and all n ∈ N. 3.4. New techniques: Quasi-Frobenius maps after Kaledin and the Z/p-equivariant product maps. As the Atiyah-Bott localization Theorem fails for the infinite-dimensional space Lφp M, the Smith inequality cannot be deduced from a spectral sequence argument associated to the /p-equivariant Floer cohomology HF ∗ (φp). In general, there is no localization map Z Z/p i∗ : HF ∗ (φp) → HF ∗(φ)⊗H∗(B /p; ), which becomes an isomorphism after inverting u (the Z/p Z Fp ∗ generator of H (BZ/p; Fp) of degree 2). Nevertheless, inspired by the work of Kaledin [22] on non-commutative geometry in characteristic p, we defined a quasi-Frobenius map from HF ∗(φ) ∗ ⊗p to the group cohomology associated to the cyclic action of Z/p to CF (φ) , whose composition with a Z/p-equivariant product map gives a map in the reverse direction P ◦ Q: HF ∗(φ) −→Q H∗( /p, CF ∗(φ)⊗p) −→P HF ∗ (φp), Z Z/p [x] 7→ [x⊗p] 7→ P(x ⊗ · · · ⊗ x). 1 Let Sp be a branched cover of R × S which has a single branched point at (0, 0) of ramification index p. The above Z/p-equivariant product map P is defined by the counts of Jz-holomorphic maps from Sp to the symplectization of the mapping torus Mφ = R × M/(t, x) ∼ (t + 1, φ(x)) Research Statement Jingyu Zhao for some domain-dependent almost complex structure Jz parametrized by Z/p-equivariant cells in S∞. Having defining these operations P and Q appropriately, one can show that their com- position gives an alternative replacement for the localization map in the reverse direction. Proposition 3.3 (Shelukhin-Z.). The composition P ◦ Q: HF ∗(φ) → HF ∗ (φp) induces an Z/p −1 ∗ ∼ 2 quasi-isomorphism after tensoring both sides by u H (BZ/p; Fp) = Fp((u))[θ]/θ . 3.5. Proposed future project: Quantum Steenrod algebra. Under additional assump- tions on M, there is a well-defined localization map i∗ constructed in [34] when p = 2 such that the quantum analogue of the total Steenrod operation on HF ∗(φ) can be obtained as the composition

∗ Sq : HF ∗(φ) −−−→P◦Q HF ∗ (φp) −→i HF ∗(φ) ⊗ H∗( P ∞) ∼ HF ∗(φ)[[h]], Z/p R = Sq(x) = x ∪ x + Sq|x|−1(x)h + Sq|x|−2(x)h2 + ··· . Such quantum Steenrod operations were first studied in the work of Fukaya and Betz-Cohen [6,9] and the recent work of N. Wilkins [37]. I am interested in constructing these operations for general prime p ≥ 3, which are the analogues of the cyclic reduced power operations defined by Steenrod. Together with the case when p = 2, this defines the quantum Steenrod algebra whose properties remain to be investigated.

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