Papers

We study torus surgery on the Cartwright–Steger surface and the degree-two cover construction proposed in \cite[Remark~1]{ASY}. Every torus in this surface is integrally nullhomologous, and surgery on a fixed disjoint collection of such tori cannot decrease the first Betti number. We then determine the nodal-curve subgroup and complete the fundamental-group calculation in the degree-two construction. The resulting minimal symplectic manifold has signature zero and fundamental group $\Z/2$; for compatible symplectic gluings, its universal cover is an exotic symplectic $31\CP^2\#31\overline{\CP}^{\,2}$. The construction on the original Cartwright–Steger surface gives a minimal symplectic manifold with signature $-1$ and fundamental group $(\Z/2)^2$.

This manuscript aims to advance existing approaches to the abelianization of symplectic quotients. Two such approaches include the imploded cross-sections of Guillemin-Jeffrey-Sjamaar, and the Gelfand-Cetlin abelianizations of the first author and Weitsman. Using our notion of generalized Gelfand-Cetlin data, we generalize the second approach to account for irregular values of the moment map. Connections to symplectic implosion are emphasized throughout this manuscript.

We show that two infinite families of homology $S^2\times S^2$'s presented in \cite{FPS, AP}, originally proposed as exotic candidates, are aspherical. In particular, none of these manifolds is an exotic $S^2\times S^2$. We prove this by reconstructing them from aspherical building blocks, following Edmonds. The same arguments apply to and rule out further candidates for simply connected exotica. We also give counterexamples to the conjecture that Luttinger surgery preserves symplectic asphericity.

Let $G$ be a Lie group equipped with commuting left and right linear representations on a vector space $V$, and let $G\Join V$ be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group $G\Join V$ from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their $\mathfrak g$- and $V$-components, where $\mathfrak g$ is the Lie algebra of $G$. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on $\mathbb R^r$, $G_r^2$. For the matrix model, the conjugation representation decomposes $\operatorname{Mat}(3)$ into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a $(1+1)$-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.

In this paper we use the topological vertex formalism to derive a structural formula for the leading terms of the generating series of Gromov–Witten invariants of local $\mathbb P^2$. Then we give two applications of this formula. First we use this result to derive an explicit formula for the leading terms of the transformed Gopakumar–Vafa invariants of local $\mathbb P^2$. This is an extension of a formula of Guo–Zhou \cite{gz} from degree $2d-5$ to $3d-1$, and shares some similar features with the famous Göttsche–Yau–Zaslow formula. And as the second application, we prove Guo–Wu–Moreira's Conjecture \cite{gw} about the leading Betti numbers of the moduli space of one-dimensional Gieseker semistable sheaves on $\mathbb P^2$.

We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.

We construct and analyze a mirror family for the generalized complete intersection Calabi-Yau threefold $X_+=[\mathbb{P}^1\,|\,3\ {-1};\ \mathbb{P}^4\,|\,2\ 3]$, with $(h^{1,1},h^{2,1})=(2,46)$. An auxiliary-variable presentation turns the negative-degree section into a variation of toric GIT. Crossing one wall flops 16 disjoint $(-1,-1)$-curves and yields a Calabi-Yau threefold $X_-$, a nef complete intersection in a toric $\mathbb{Q}$-Fano eightfold, which carries a genus-one fibration of index 4. Because this ambient space is not Gorenstein, the Batyrev-Borisov construction does not apply. We use the Hori-Vafa equations only to select a 2-parameter Laurent family, and construct a smooth projective crepant compactification $Y$ of the Laurent model with $(h^{1,1},h^{2,1})=(46,2)$; for very general parameters the Mordell-Weil group is $\mathbb{Z}/4\mathbb{Z}$. The periods of $Y$ satisfy a rank-6 Picard-Fuchs system with two maximally unipotent boundary points, whose indicial algebras are isomorphic to the rational even cohomology rings of $X_-$ and $X_+$. The resulting genus-zero predictions agree with independent counts of 176 vertical lines and 100 vertical conics on the genus-one fibration $X_-\to\mathbb{P}^2$.

The configuration matrices of generalized complete intersection Calabi-Yau (gCICY) threefolds contain negative entries, so their defining line bundles are not nef and the Batyrev-Borisov construction does not apply directly. We study gCICY threefolds in $\mathbb{P}^1\times B$ with a $[3,-1]$ block, that is, cut out by sections of $O(3)\boxtimes L$ and $O(-1)\boxtimes(L+R)$, where $B$ is a smooth projective fourfold and $2L+R=-K_B$. Auxiliary variables turn the generalized section into a relative variation of GIT, whose opposite phase is the zero locus of a rank-3 bundle on $\mathbb{P}^2\times B$; for general data the two phases are isomorphic or related by $\int_B c_1(L)^4$ disjoint Atiyah flops. Over products of projective spaces there are exactly 18 Calabi-Yau threefold configurations with a $[3,-1]$ block. All of them except the non-Gorenstein case studied in the companion paper have Gorenstein toric nef phases. Thus, for every other configuration in this classification, mirror symmetry is reduced after variation of GIT to ordinary Batyrev-Borisov duality, which gives the mirror Hodge numbers of the generalized phase itself. We analyze four examples with a $[3,-1]$ block and one outside this class in detail. For each we construct an explicit mirror and compute its fundamental period and low-order Picard-Fuchs system; the indicial algebras at the maximally unipotent boundary points are isomorphic to the rational even cohomology rings of the A-model phases.

To an oriented quiver with boundary $\Gamma$ and a compact connected Lie group $G$ we associate a Nahm-Kirchhoff moduli space $\mathcal{M}_{\mathbb{H}}(\Gamma)$: solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected $\Gamma$, $\mathcal{M}_{\mathbb{H}}(\Gamma)$ is a smooth manifold of dimension $4(|E|-|\Gamma_{\mathrm{int}}|)\dim G$ with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold $G_{\mathbb{C}}/G$, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient $\mathcal{M}_{\mathbb{C}}(\Gamma)$ of the complex Nahm equations by the complexified gauge group. This quotient is independent of edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^*G_{\mathbb{C}})^E$, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, $\Gamma\mapsto\mathcal{M}_{\mathbb{C}}(\Gamma)$ is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on $\mathbf{Cob}_2$: spines of a cylinder with different lengths give non-isomorphic spaces. Retaining lengths, these spaces form a metric field theory, a functor of non-unital monoidal categories from metric cobordisms to hyperkähler manifolds composed by hyperkähler reduction; over each cone of $M_{g,n}^{\mathrm{trop}}$ they give a family with fixed complex-symplectic fibre and varying metric. Finally, harmonic gauge representatives converge as an internal edge collapses, and on a long edge with regular semisimple Higgs field the Kempf-Ness minimizer is exponentially confined near its centralizer; the limiting metrics are left open.

We construct smooth time-periodic Hamiltonians on the monotone manifolds \(S^2\times S^2\) and \(\CP^2\) for which the Clifford torus meets its Hamiltonian image in exactly two isolated degenerate points.

It is shown that a symplectic isotopy with non-degenerate end-point on a closed symplectically Calabi–Yau manifold has simple periodic points of arbitrarily large period under a simple non-vanishing condition on the Morse–Novikov homology associated with its flux. The proof combines a sharp estimate for the Conley–Zehnder index under the assumption that the mean index is an even integer with a classical asymptotic iteration argument in the non-Hamiltonian Floer-theoretic setting. We also apply this estimate to the symplectic dynamics of the standard symplectic $4$-torus.

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

We show that not every symplectic Lefschetz fibration over the $2$–sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

We define a symplectic structure on singular 1-form densities and a presymplectic structure on augmented triples, a compressible analog of the Marsden–Weinstein symplectic structure on vortex membranes in the incompressible case. Equivalence classes of augmented triples can be regarded as special vorticities in compressible fluids, i.e., singular elements of the dual to the Lie algebra of the group of all diffeomorphisms of a manifold, while Marsden–Weinstein symplectic structure is related to singular vorticities for the group of volume-preserving diffeomorphisms. This structure on singular 1-form densities occupies an intermediate position between two other settings, which we revisit, the Weinstein symplectic structure on weighted isotropic submanifolds of symplectic manifolds and the Marsden–Weinstein symplectic structure on vortex membranes.

Let $(M,\omega)$ be a closed monotone integral symplectic manifold with monotonicity constant $\tau>0$, and let $D\subset M$ be a symplectic hyperplane section of degree $0<k<\tau$. By Biran's decomposition theorem, $M$ is the union of a symplectic disk bundle over $D$ and the skeleton of the complement $M\setminus D$ equipped with its Liouville structure. In this paper, we compute the relative symplectic cohomology of circle subbundles of this disk bundle over $\mathbb{Z}$ in terms of the quantum cohomology of $D$. This relative symplectic cohomology depends on the radius of the circle bundle relative to a certain critical radius. For a broad class of pairs $(M,D)$, we prove that the relative symplectic cohomology over $\mathbb{Z}$ does not vanish for supercritical radii. Therefore, circle bundles of supercritical radii are heavy subsets, which yields infinite dimensional quasi-flats in the Hamiltonian diffeomorphism group with respect to both the Hofer metric and the spectral metric over $\mathbb{Z}$. We also establish that if the degree $k$ of $D$ is greater than one, the skeleton of $M\setminus D$ is SH-full over $\mathbb{Z}/k\mathbb{Z}$, and hence non-displaceable in $M$ by any symplectomorphism.

Given a polygon in the plane with vertices on lattice points, Pick's theorem expresses its area A in terms of the number I of lattice points in its interior and the number B of lattice points in its boundary: A = I + B/2 - 1. We consider polygons in the plane whose edges are halfway between lattice points. For such a polygon that is also unimodular, we express its area A in terms of the number I of lattice points in its interior and the number V of its vertices: A = I - V/8 + 1/2 .

Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to $\CP^2\#k\CPb^2$, $k=3,5$ \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a $k=5$ example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.

A symplectic spectral function of a $2n \times 2n$ real positive definite matrix $A$ is a function of its symplectic eigenvalues $0< d_1(A) \leq \cdots \leq d_n(A)$, which is given by a composition $f \circ d$ of some symmetric function $f$ on the set of $n$-vectors with positive entries and the symplectic eigenvalue vector map $d(A)=(d_1(A),\ldots, d_n(A))$. In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function $f \circ d$ is Fréchet differentiable at $A$ if and only if $f$ is Fréchet differentiable at $d(A)$, and we compute the derivative expression explicitly. We also show that $f \circ d$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $A$ if and only if $f$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $d(A)$. We determine the Clarke generalized gradient of a symplectic spectral function $f \circ d$ at $A$, which is given in terms of the Clarke generalized gradient of $f$ at $d(A)$. As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.

We show that circle-valued spectral values in Tate Rabinowitz Floer homology give rise to quantized energy levels, which recover the quantum mechanical spectrum of a system of uncoupled harmonic oscillators.

We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in $\mathbb{P}^3$ or sextics in $\mathbb{P}(3,1,1,1)$ and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.

A classical 1977 result by Roter states that a Riemannian manifold of dimension at least four with parallel Weyl curvature must be locally symmetric or conformally flat. We establish Roter's theorem in the symplectic category: namely, a Fedosov manifold with parallel symplectic Weyl curvature must be locally symmetric or Weyl-flat.

We find explicit formulas for the EHZ and cylindrical capacities of rotated $L_{p}$-ellipsoids. In addition, we provide an algorithm to estimate the EHZ capacity.

Let $\Sigma_g$ be a closed oriented surface of genus $g\geq2$, and let $\mathscr F_g$ be its split-closed, strictly unobstructed, two-periodic Fukaya category over the complex Novikov field $\Lambda$. We determine the derived Picard group \[ \operatorname{DPic}_{\Lambda}(\mathscr F_g) \cong \bigl(H^1(\Sigma_g;\Lambda^\times) \rtimes\pi_0\operatorname{Diff}^+(\Sigma_g)\bigr) \times\mathbb Z/2\mathbb Z. \] This proves the enhanced form of the conjecture of arXiv:2006.09689. As a corollary, we obtain that the derived Picard group itself determines the genus of the surface.

While structure-preserving methods for the forward simulation of Hamiltonian systems are well established, the mathematical foundations of their data assimilation and filtering counterparts remain comparatively less developed, particularly regarding rigorous accuracy guarantees. To address this gap, we propose an online filtering algorithm to reconstruct an unknown function $u^\dagger$ from finitely many measurements, assuming that $u^\dagger$ solves a parametric Hamiltonian PDE with unknown inputs. The method uses low-dimensional symplectic approximation spaces that evolve in time according to the PDE model and informed by the measurement data. The method can also be coupled with a dynamical sensor-placement strategy designed to optimize reconstruction stability. We derive error bounds in terms of stability and best approximation, and show that the reconstruction remains in a symplectic space and preserves energy up to the approximation error. Numerical experiments demonstrate the effectiveness of the method and the benefits of dynamic over static sensor placement.

We associate to each compact subset $K$ of a closed symplectic manifold $M$ an $A_\infty$-category $\Fuk_K$, called the Fukaya category with support on $K$. The objects are tautologically unobstructed global Lagrangians endowed with appropriate decorations. The dependence on $K$ is detected by the morphism complexes which can be computed by completed telescopes associated with acceleration data for $K$. In particular, these vanish for Lagrangians disjoint of $K$. For $K=M$ this recovers the usual Fukaya category of tautologically unobstructed objects. These categories admit restriction functors under inclusions of compact subsets. We prove a Mayer–Vietoris descent theorem for weakly involutive covers. The construction is carried out over the Novikov ring and is established whenever classical transversality methods apply.

The existence of orbifold structures on mapping spaces between orbifolds was established in previous work, notably by W. Chen and B. Chen-Du-Liao. In this paper, we construct an infinite-dimensional orbifold structure on the mapping space in a more geometric way, using bibundles (or Hilsum-Skandalis morphisms), based on Lerman's description of orbifold morphisms. We construct a proper étale (or Fréchet) Lie groupoid representing the mapping space in the $C^k$, $C^\infty$, and $W^{k,p}$ settings, and show that its Morita equivalence class is independent of the choices of groupoid presentations. Locally, the mapping groupoid is isomorphic to the action groupoid of a finite group, where the finite group is identified with the automorphism group of the corresponding bibundle. This provides a concrete geometric framework for further studies of mapping spaces and moduli spaces on orbifolds, with applications to Floer theory on symplectic orbifolds.

The Katok examples on $S^2$ are induced by Randers metrics obtained by perturbing the round metric by the standard rotational Killing field: allowing a scaling factor of the round metric gives a two-parameter family of Randers metrics $F_{\alpha,\beta}$, parametrised by positive real numbers $(\alpha,\beta)$. We compute the set of all $(\alpha,\beta)\in (0,2+\sqrt{3})^2$ for which $D^*(S^2,F^*_{\alpha,\beta})$, the unit codisc bundle with respect to $F^*_{\alpha,\beta}$, symplectically embeds into the round codisc bundle $D^*S^2$. This set has the structure of an infinite staircase. We also establish a dictionary between embeddings of these codisc bundles, singular $A_1$-ellipsoid embeddings, and $\mathbb{Z} _2$-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the $\mathbb{R} P^2$ case and discuss a broader family of examples for which this dictionary applies.

We establish the use of Lagrangian diagram moves to build Lagrangian cobordisms of Legendrian knots. We then apply the moves to explore how to invert decomposable cobordisms, after adding some stabilizations. This is done by use of Reidemeister moves and handle attachments in the Lagrangian projection. As an application, we show an explicit construction of a Lagrangian concordance from a stabilized $m(9_{46})$ Legendrian knot to a stabilized unknot, i.e. a non-decomposable Lagrangian concordance of nonempty knots.

We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every $\alpha\geq 0$ there is a sequence of Hamiltonian diffeomorphisms $\phi_k$ on the two-torus $T^2$ such that $$d_H(\operatorname{Ent}_{\leq \alpha}(T^2,\omega),\phi_k)\to \infty \quad (k \to \infty),$$ where $\text{Ent}_{\leq \alpha}(T^2,\omega)\subset \text{Ham}(T^2,\omega)$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $\alpha$ and $d_H$ the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.