Given a Hamiltonian torus action on a symplectic manifold, Teleman and Fukaya have proposed that the Fukaya category of each symplectic quotient should be equivalent to an equivariant Fukaya category of the original manifold. We lay out new conjectures that extend this story – in certain situations – to singular values of the moment map. These include a proposal for how, in some cases, we can recover the non-equivariant Fukaya category of the original manifold starting from data on the quotient.

To justify our conjectures, we pass through the mirror and work out numerous examples, using well-established heuristics in toric mirror symmetry. We also discuss the algebraic and categorical structures that underlie our story.

We construct moduli spaces of framed logarithmic connections and also moduli spaces of framed parabolic connections. It is shown that these moduli spaces possess a natural algebraic symplectic structure. We also give an upper bound of the transcendence degree of the algebra of regular functions on the moduli space of parabolic connections.

In this paper, we establish homological Berglund–Hübsch mirror symmetry for curve singularities where the A–model incorporates equivariance, otherwise known as homological Berglund–Hübsch–Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda which posits that the equivariance in the A–model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B–model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the Fan–Jarvis–Ruan–Witten (FJRW) A–model, a result which might be of independent interest for its implications in the Landau–Ginzburg analogue of Dubrovin’s conjecture.

Let X be a toric Calabi-Yau 3-fold and let $L\subset X$ be an Aganagic-Vafa outer brane. We prove two versions of open WDVV equations for the open Gromov-Witten theory of $(X,L)$. The first version of the open WDVV equation leads to the construction of a semi-simple (formal) Frobenius manifold, and the second version leads to the construction of a flat (formal) F-manifold.

We prove a genus zero Givental-style mirror theorem for all complete intersections in toric Deligne-Mumford stacks, which provides an explicit slice called big I-function on Givental’s Lagrangian cone for such targets. In particular, we remove a technical assumption called convexity needed in the previous mirror theorem for such complete intersections. In the realm of quasimap theory, our mirror theorem can be viewed as solving the quasimap wall-crossing conjecture for big I-function [13] for these targets. In the proof, we discover a new recursive characterization of the slice on Givental’s Lagrangian cone, which may be of self-independent interests.

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

We compute the open Gromov-Witten disk invariants and the relative quantum cohomology of the Chiang Lagrangian $L_\triangle \subset \mathbb {C}P^3$. Since $L_\triangle $ is not fixed by any anti-symplectic involution, the invariants may augment straightforward J-holomorphic disk counts with correction terms arising from the formalism of Fukaya $A_\infty $-algebras and bounding cochains. These correction terms are shown in fact to be nontrivial for many invariants. Moreover, examples of nonvanishing mixed disk and sphere invariants are obtained.

We characterize a class of open Gromov-Witten invariants, called basic, which coincide with straightforward counts of J-holomorphic disks. Basic invariants for the Chiang Lagrangian are computed using the theory of axial disks developed by Evans-Lekili and Smith in the context of Floer cohomology. The open WDVV equations give recursive relations which determine all invariants from the basic ones. The denominators of all invariants are observed to be powers of $2$ indicating a nontrivial arithmetic structure of the open WDVV equations. The magnitude of invariants is not monotonically increasing with degree. Periodic behavior is observed with periods $8$ and $16.$

We provide two constructions of Gaussian random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta )$, that are particularly interesting in the case where the space of $\mathcal {L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if $H^{0}_{(2)}(X,L)$ is infinite dimensional, are almost never $\mathcal {L}^2$-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin–Toeplitz quantization and yields a Gaussian ensemble of random $\mathcal {L}^2$-holomorphic sections. Furthermore, we study their random zeros in the context of semiclassical limits, including their distributions, large deviation estimates, local fluctuations and hole probabilities.

Let W be a symplectic manifold, and let $\phi :W \to W$ be a symplectic automorphism. This automorphism induces an auto-equivalence $\Phi $ defined on the Fukaya category of W. In this paper, we prove that the categorical entropy of $\Phi $ provides a lower bound for the topological entropy of $\phi $, where W is a Weinstein manifold and $\phi $ is compactly supported. Furthermore, motivated by [cCGG24], we propose a conjecture that generalizes the result of [New88, Prz80, Yom87].

In this paper we use the periodic Toda lattice to show that certain Lagrangian product configurations in the classical phase space are symplectically equivalent to toric domains. In particular, we prove that the Lagrangian product of a certain simplex and the Voronoi cell of the root lattice $A_n$ is symplectically equivalent to a Euclidean ball. As a consequence, we deduce that the Lagrangian product of an equilateral triangle and a regular hexagon is symplectomorphic to a Euclidean ball in dimension 4.

We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with singular fibers beyond the topological level. The dual singular fibration is explicitly written and proved to be compatible with the family Floer mirror construction. Moreover, we discover that the Maurer-Cartan set of a singular Lagrangian is only a strict subset of the corresponding dual singular fiber. This responds negatively to the previous expectation and leads to new perspectives of SYZ singularities. As extra evidence, we also check some computations for a well-known folklore conjecture for the Landau-Ginzburg model.

We show that the image of a properly embedded Legendrian submanifold under a homeomorphism that is the $C^0$-limit of a sequence of contactomorphisms supported in some fixed compact subset is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any closed non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen–Zhang result on the degeneracy of the Chekanov–Hofer–Shelukhin pseudo-norm for properly embedded non-Legendrians.

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma $ that we construct are either Birkhoff sections, which means that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial \Sigma $ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma $. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.

Let $X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$ or, equivalently, the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional ‘affine $A_1$-case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.

If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$-type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

A complete embedding is a symplectic embedding $\iota :Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota (K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.