Papers

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. It is called $\epsilon$-positive if the ratio of the restrictions of the imaginary and real parts of the holomorphic volume form is bounded by $\frac{1}{\epsilon}.$ An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We prove that the Riemannian metric restricted to the subset of $\epsilon$-positive Lagrangians induces a path metric when the relative Kähler class of the ambient Calabi-Yau manifold lies on a line of rational slope. The proof relies on the construction of a sequence of quantization maps from a circle bundle over the space of positive Lagrangians to the dual of the space of holomorphic sections of powers of a prequantum line bundle. The rationality condition for the relative Kähler class allows us to reduce to the case of Bohr-Sommerfeld Lagrangians. Building on ideas of Tian and Nirenberg-Wells, Hörmander estimates are used to show that functions on a Bohr-Sommerfeld Lagrangian can be approximated by holomorphic sections of a sufficiently high power of the prequantum line bundle. Consequently, the quantization maps eventually separate fibers of the circle bundle.

Generalized complex structures give rise to generalized Dolbeault, Bott–Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0\ltimes_A\mathfrak h$, where $\mathfrak h$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $\Gamma\backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley–Eilenberg cohomology of the complex Lie algebra $\mathcal L$, the $+{\mathbf i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal L$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb R$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $\Gamma\backslash G_A$.

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $\Lambda\subset(\mathbb{S}^{2n+1},\xi_{\mathrm{std}})$ with \[ \mathrm{tb}(\Lambda)\ne (-1)^{\frac{n(n-1)}{2}+1} \] has vanishing contact homology. In dimension three, this produces new examples of tight algebraically overtwisted contact manifolds, vastly extending the authors' previous results: If a Legendrian knot in $(\mathbb{S}^3,\xi_{\mathrm{std}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1/k)$-surgery on it is algebraically overtwisted and tight for every positive integer $k$. Our proofs use versions of symplectic cohomology and wrapped Floer cohomology defined by augmentations of a chain-level contact homology algebra.

We develop a homological model for hamiltonian reduction of Courant algebroids, parallel to the BFV formulation of hamiltonian reduction in symplectic geometry. To a hamiltonian action on a Courant algebroid, we associate a differential graded Courant–Dorfman algebra, obtained from the BFV algebra of the corresponding graded symplectic manifold; under usual regularity assumptions, its cohomology recovers the Courant–Dorfman algebra of the reduced Courant algebroid.

We study the irrationality of smooth Fano complete intersection fourfolds in weighted projective spaces, using an atom theoretic approach based solely on Gromov–Witten invariants. We prove symplectic irrationality, and hence irrationality for every smooth member, in a wide range of families for which irrationality was previously known only for very general members.

We show that every symplectic manifold admits a symplectic form whose metaplectic-c prequantizations give quantizability conditions on all its Lagrangian submanifolds. Metaplectic-c prequantizations descend to prequantizations over the Lagrangian Grassmannian bundle that are fibrewise flat with holonomies primitive fourth roots of unity on the generators of the fundamental groups. This fibrewise holonomy is consistently interpreted as that of both Leray two-orientations and normalized Blattner–Kostant–Sternberg half-forms. The prequantizations pull back to flat circle bundles on each Lagrangian submanifold whose holonomy character defines a quantizability condition. The first and second Stiefel–Whitney classes are identified as splitting obstructions to the Lagrangianwise holonomies. These holonomies split into a product of holonomies on any orientable Lagrangian submanifold, and split consistently for all Lagrangian submanifolds of a manifold with metaplectic structure. When the first Chern class vanishes, the holonomy splits into the Bohr–Sommerfeld factor and a quarter phase Maslov correction, recovering the classical Einstein–Brillouin–Keller condition. We develop the underlying obstruction theory for circle extensions associated to general Lie group double coverings with first hypercohomology groupoids.

We study the recovery of a Hamiltonian from discrete trajectory observations using structure-preserving kernel methods. When exact-flow data are fitted through a symplectic one-step scheme, the fixed-step learning target is generally not the physical Hamiltonian itself, but a scheme-dependent inverse modified Hamiltonian. For the implicit midpoint and $p$-implicit symplectic Euler methods, we establish the existence of analytic inverse targets and show that their discrepancies from the physical Hamiltonian are second and first order in the observation step, respectively. Within an RKHS framework, we derive finite-sample reconstruction bounds for random trajectories, allowing dependence among observations from the same trajectory, and separate the error into estimation, approximation, and integrator-induced components. We further establish stability under noisy trajectory observations and develop a multistride Richardson correction that systematically raises the order of the integrator-induced error without requiring additional trajectories. Numerical experiments confirm the predicted scheme-dependent orders, the effect of Richardson correction, and the applicability of the method across several Hamiltonian systems.

This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.

We construct closed symplectic 4-manifolds admitting the structure of hyperbolic $\pi$-orbifold, addressing LeBrun's conjecture. Our simplest example has Euler characteristic $304$, is tessellated by $64$ copies of the right-angled hyperbolic $120$-cell and admits a four-fold branched covering over a product of two surfaces of genus $4$.

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

This Master's thesis develops and applies the multisymplectic field-theory framework to (pure) Yang-Mills theory. Building from the geometric foundations of jet bundles and the Poincaré-Cartan forms, the work constructs the Lagrangian multisymplectic formalism for first-order field theories, recovering the usual Euler-Lagrange field equations and extending the Poisson brackets of symplectic mechanics to a full graded Lie algebra. It then establishes the Legendre correspondence with the multisymplectic Hamiltonian formalism (in which Hamilton equations are retrieved) including the case of singular Lagrangians (where a constraint algorithm is reproduced). Breaking Lorentz covariance via a Cauchy surface slicing, the thesis then relates the covariant multisymplectic picture to an instantaneous presymplectic (infinite-dimensional) formulation in which the instantaneous De Donder equations and presymplectic brackets are obtained in a compatible way. A comprehensive treatment of symmetries then leads to multisymplectic and instantaneous momentum maps generalizations and to several versions of Noether's theorems. Specializing the formalism to principal-bundle gauge theory, the thesis identifies the multisymplectic configuration space with the bundle of connections, gauge symmetries with symmetries in the multisymplectic language and builds the Yang-Mills Lagrangian. Lastly, among many computations, the standard tensorial Yang-Mills equations and the Bianchi identity in local trivializations are retrieved, Yang-Mills conserved currents and charges are obtained and the graded algebra is derived for a special class of Hamiltonian forms.

We give a complete classification of tight contact structures on manifolds obtained from any surgery along any torus knot, which provides the first such classification on an infinite family of knots. The main tools are the classification of non-loose torus knots, bordered contact invariants via their immersed curve interpretations, and non-thickenable tori in knot complements. As a byproduct of this classification, we also give the first classification of contact structures on some surface bundles over the circle with fibers of genus larger than $1$. The symplectic fillability of these contact structures is also studied and leads to, among other things, the disproof of a conjecture of Ghiggini and Van Horn-Morris on Stein fillability of some contact structures on a Brieskorn sphere.

The main result is an obstruction to the presence of contact regions for c-symplectic confoliations in any odd ambient dimension. As a first application, we prove a 5-dimensional analogue (under stronger assumptions) of a Reeb stability theorem for confoliations in dimension 3 due to Eliashberg and Thurston. Refining work of the first author with Cardona, we further give two new notions of contact approximations of c-symplectic confoliations in any odd ambient dimension generalizing the 3-dimensional notion of Eliashberg-Thurston. As a second application of our main theorem, we then give an example, in any ambient odd dimension, of a c-symplectic (and in fact strong symplectic) foliation that cannot be contact approximated, in either of these two senses. This relies on the property, of independent interest, that contact structures approximating (in these senses) a weakly c-symplectic confoliation are also weakly fillable. The proofs of all these statements rely on pseudo-holomorphic curve theory. Lastly, we describe several natural classes of examples of these types of contact approximations of c-symplectic confoliations.

We show that the cotangent bundle of a holomorphically flexible Stein manifold has the symplectic density property. As application we deduce holomorphic approximation for Hamiltonian diffeomorphisms of real cotangent bundles by complex symplectic automorphisms of the complexified cotangent bundles, which preserve the real bundle, in the Carleman sense, i.e. in the Whitney topology.

We derive the noncommutative open WDVV equations for the open Gromov-Witten superpotential introduced in joint work of the author and Solomon, extending the Solomon-Tukachinsky equations to noncommutative differential graded algebras. The resulting recursive formulas facilitate the computation of open Gromov-Witten invariants with multiple boundary point constraints for even-dimensional Lagrangians. We apply the open WDVV equations to compute all open Gromov-Witten invariants for the real loci of even-dimensional complex quadrics of dimension greater than two. These computations show that open Gromov-Witten invariants with multiple boundary point constraints can be nontrivial in arbitrary dimensions. We also construct a character for a natural subalgebra of big quantum cohomology associated to a Lagrangian endowed with a point-like bounding cochain. The character is in the spirit of the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono. Applying the character construction to the even-dimensional quadric illustrates how point-like bounding cochains refine the information about closed quantum cohomology obtained from weak bounding cochains of Floer theory. Open Gromov-Witten invariants with three boundary point constraints determine the two extensions of the character from the subalgebra to big quantum cohomology.

We solve a well-known question of Polterovich from 2002 regarding the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two: every non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterations separated from the identity in Hofer's metric. In higher dimensions, we make progress on the case of autonomous Hamiltonian flows: we prove Hofer non-recurrence for symplectically hyperbolic manifolds, and completely characterize Hofer recurrence for functions of the moment polytope on monotone toric manifolds. These results apply equally well to Viterbo's spectral metric, settling cases of the $\gamma$-rigidity conjecture, and therefore in many cases also to the $C^0$-metric. Our approach relies on the philosophy that symplectically visible invariant sets govern symplectic non-recurrence phenomena. In dimension two, we use invariant annuli provided by low-dimensional dynamics, and apply methods of $C^0$ symplectic topology to approximately invariant Lagrangian submanifolds. For symplectically hyperbolic manifolds, the invariant sets are provided by sublevel and superlevel sets of autonomous Hamiltonians, while in the toric case, they comprise Lagrangian torus fibers. Across the arguments, we use quantitative Lagrangian Floer theory and its relation to notions of classical dynamics.

We discuss the recently introduced concept of hyperquivers, due to Muller, Nanda, and Seigal, from the point of view of Kähler and hyperkähler quotients. In particular, we show that the Kähler hyperquiver variety ${\mathcal M}$, corresponding to a hyperquiver with a single hyperedge, is a projective variety isomorphic to the geometric quotient of the space of tensors with maximal multilinear rank. Moreover, the hyperkähler hyperquiver variety corresponding to doubling of this hyperquiver can be identified with the linear scheme corresponding to the sheaf of reflexive $1$-forms on ${\mathcal M}$.

We study several fundamental properties of the category $\mathbf{Kur}$ introduced in \cite{Kim1}. By defining fibrations and weak equivalences, we show that morphisms with manifold targets admit homotopy fiber products; this is then applied to obtain a notion of generalized intersections of two submanifolds. A definition of the Weinstein category in the context of Kuranishi spaces is also presented. Finally, we establish that $\mathbf{Kur}$ forms a category of fibrant objects in the sense of Brown \cite{Brown}, and discuss its homotopy category $\operatorname{Ho}(\mathbf{Kur})$.

Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge2$. Let $\mathcal T_MG$ be one of the following transformation groups: $Homeo_0(M,\mu)$, $Diff_0(M,\mu)$, $Symp_0(M, \omega)$ (in case $M$ is symplectic) and $Ham(M, \omega )$ (in case $M$ is symplectic). Denote by $\overline{H}_b^d(\mathcal T_M)$ reduced bounded cohomology of $\mathcal T_M$ in degree $d$, and by $\overline{EH}^d(\mathcal T_M)$ reduced exact bounded cohomology of $\mathcal T_M$ in degree $d$. In this paper we prove that if $n=2$ and $M$ is a closed surface $\Sigma_g$ or a disc $\mathbb D$, then for every $d\ge 2$ $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, $\dim\overline{EH}^d(\mathcal T_M)=\infty$ if $\mathcal T_M$ is either $Diff_0(\Sigma_g,\mu)$ or $Ham(\Sigma_g, \omega)$, or $Ham(\mathbb{D}, \omega)$; or $g>1$. In case $n>2$, under certain conditions on $\pi_1(M)$, we prove that for every degree $d\ge 2$ $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for $\mathcal T_M$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

We develop a theory of bi-contact plugs on oriented $3$-manifolds to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a hyperbolic plug with filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted unoriented contact form. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic projectively Anosov flows without saddle periodic orbits and Morse-Smale examples with prescribed saddle count. Generalizing the construction of Bonatti-Bowden-Potrie, finite capping embeds any bi-contact plug in a closed projectively Anosov flow with the same topological entropy, preserving partial hyperbolicity for attracting plugs. We also construct flows with controlled invariant torus dynamics on torus bundles and on doubles of plugs with coorientable boundary bi-foliations. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

We show that classical cylindrical contact homology over $\mathbb{Z}$, equipped with its decomposition by free homotopy classes, is not invariant under changes of hypertight contact form. Specifically, we construct two nondegenerate hypertight perturbations of the standard Seifert contact form on $\Sigma(2,4,5)$ whose cylindrical contact homologies in the regular-fiber class are respectively torsion-free and contain $\mathbb{Z}/5$-torsion.

We investigate the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, with particular emphasis on cotangent bundles of generalized flag manifolds. Our approach gives an explicit description of the Liouville symplectic form on these spaces in terms of the connections and curvature forms of principal bundles naturally arising from the Lie group structure. This provides a geometric link between the Lie-theoretic properties of generalized flag manifolds and the symplectic structures carried by their cotangent bundles.

We introduce, for an embedded Legendrian link~$L$ in a contact 3-manifold~$(M,\xi)$, the group of its Legendrian motions. Analogously with the smooth case, we prove that this Legendrian motion group is isomorphic to the fundamental group of the space of unparametrised Legendrian embeddings. For all Legendrian knots in the standard contact 3-sphere, we characterise Legendrian motions in terms of contact mapping classes of the knot exterior. In an appendix, we give a short exposition of our techniques applied to the smooth case, comparing to the classical Dahm homomorphism.

We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston–Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.

To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a $1$-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated $0$-shifted Poisson structure. However, such morphisms form only a special class of $1$-shifted Lagrangians. We extend this picture to $1$-shifted Lagrangians and, more generally, $1$-shifted coisotropics in $1$-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo–Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.

We study exact fillings of Legendrians and contact manifolds using Floer theory of universal covers. We prove vanishing results on wrapped Floer homology of the universal cover and symplectic (co)homology with $k[\pi_1]$ coefficients by using lifted versions of Ritter's TQFT operations. We use these results to show that an exact Maslov zero Lagrangian filling of the standard Legendrian $\Lambda_{std}$ in any Liouville filling of $(S^{2n-1},\xi_{std})$ is diffeomorphic to a ball in dimension at least 5. We show that the existence of a topological simple exact filling $W$ with vanishing symplectic homology for a dynamically convex contact manifold $M^{2n-1}$ reveals information about homotopy groups of $W$ and $M.$ We also strengthen a theorem of Zhou for ADC-contact manifolds as an application of universal covers.

We study Stein structures on nearly Lefschetz fibrations, with applications to symplectic and Lagrangian submanifolds in Weinstein domains. We prove that, up to Weinstein deformation equivalence, all Lagrangian disks with Legendrian boundary in $4$-dimensional Weinstein domains are regular in the sense of Eliashberg-Ganatra-Lazarev. This settles a Weinstein analogue of the nearby Lagrangian conjecture for two-dimensional Lagrangian disks, and resolves part of a Lagrangian analogue of the Slice-Ribbon conjecture. Along the way, we show that positive allowable nearly Lefschetz fibrations are supported by canonical Stein structures and prove a corresponding quasiflexibility result in the planar case. This yields a generalization of work of Boileau-Orevkov which may be of independent interest. Finally, we give an explicit algorithm producing a Weinstein Kirby diagram from the nearly Lefschetz fibration structure of a multisection complement.

In Beyond ECH Capacities, Hutchings used the ECH cobordism map to produce obstructions for embedding one convex toric domain into another. We generalize the methods there to obstruct symplectic embeddings of a disjoint union of convex toric domains into another convex toric domain. We also find additional obstructions coming from intersection positivity and use similar methods to obstruct the existence of isotopies between two given symplectic embeddings.

This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

We extract rank and regularized exponents from cyclic generic factors of the quantum connection and show that these data persist under the birational comparisons considered here. Applications include showing non-rationality of cubic threefolds, ordinary Gushel–Mukai threefolds, very general quartic fourfolds, and products of the threefolds with $\PP^1$.