We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.

The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.

We prove an extension of the homology version of the Hofer–Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere.

Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).

In this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.

We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated expansion rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. We then study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics, and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving, in terms of those theories. We finally use our study to show that the bi-contact surgery operations of Salmoiraghi [Surgery on Anosov flows using bi-contact geometry. Preprint, 2021, arXiv:2104.07109; and Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328] can be applied in an arbitrary small neighborhood of a periodic orbit of any Anosov flow. In particular, we conclude that the Goodman surgery of Anosov flows can be performed using the bi-contact surgery operations of Salmoiraghi [Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328].

It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on $\mathbb {P}^1$ by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver $\Gamma _n$ on two vertices and n equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver $\Gamma _n$. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver $\Gamma _n$ such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver $\Gamma _1$ by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to $\Gamma _n$ admits a factorisation in terms of n copies of the algebra attached to $\Gamma _1$.

For subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ whose closures are intersecting with coisotropic subspace $\mathbb {R}^{n,k}$ we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to $\mathbb {R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb {R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal {S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal {S}$.

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb {A}_{k}^{1}$, the derived critical locus has a natural Lagrangian fibration $\textbf {Crit}(f) \rightarrow X$. In the case where f is nondegenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective $C^\infty $ orbifold that admits a presentation $[(S^1\times \mathbb {R} ^2)/G_\tau ]$, where $G_\tau $ is a finite abelian group containing K and $G_\tau /K \cong \boldsymbol {\mu }_{\mathfrak {m}}$ is cyclic of some order $\mathfrak {m}\in \mathbb {Z} _{>0}$.

1. 1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack $\mathcal {X}$ with boundaries mapped into an Aganagic-Vafa brane $\mathcal {L}$. All genus open-closed Gromov-Witten invariants of $\mathcal {X}$ relative to $\mathcal {L}$ are defined by torus localisation and depend on the choice of a framing $f\in \mathbb {Z} $ of $\mathcal {L}$.

2. 2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure.

3. 3. When $\mathcal {X}$ is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that $G_\tau =\boldsymbol {\mu }_{\mathfrak {m}}$, we define generating functions $F_{g,h}^{\mathcal {X},(\mathcal {L},f)}$ of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )^{\otimes h}$, where $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )\cong \mathbb {C} ^{\mathfrak {m}}$ is the Chen-Ruan orbifold cohomology of the classifying space $\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}$ of $\boldsymbol {\mu }_{\mathfrak {m}}$.

4. 4. We prove an open mirror theorem that relates the generating function $F_{0,1}^{\mathcal {X},(\mathcal {L},f)}$ of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of $\mathcal {X}$. This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.

This short paper shows a topological obstruction of the existence of certain Lagrangian submanifolds in symplectic $4m$-manifolds.

We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$, where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$, of symplectomorphisms that act trivially on homology, with $k \leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $, the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$-spaces, and the computation of Seidel elements of some circle actions.

We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.

In this paper, we introduce the notion of planar two-center Stark–Zeeman systems and define four $J^{+}$-like invariants for their periodic orbits. The construction is based on a previous construction for a planar one-center Stark–Zeeman system in [K. Cieliebak, U. Frauenfelder and O. van Koert. Periodic orbits in the restricted three-body problem and Arnold’s $J^+$-invariant. Regul. Chaotic Dyn. 22(4) (2017), 408–434] as well as Levi-Civita and Birkhoff regularizations. We analyze the relationship among these invariants and show that they are largely independent, based on a new construction called interior connected sum.

We define a graph encoding the structure of contact surgery on contact $3$-manifolds and analyse its basic properties and some of its interesting subgraphs.

This paper studies a class of singular fibrations, called self-crossing boundary fibrations, which play an important role in semitoric and generalized complex geometry. These singular fibrations can be conveniently described using the language of Lie algebroids. We will show how these fibrations arise from (nonfree) torus actions, and how to use them to construct and better understand self-crossing stable generalized complex four-manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these types of fibrations (a so-called nodal trade).

Using crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

The $\rho $-Einstein soliton is a self-similar solution of the Ricci–Bourguignon flow, which includes or relates to some famous geometric solitons, for example, the Ricci soliton and the Yamabe soliton, and so on. This paper deals with the study of $\rho $-Einstein solitons on Sasakian manifolds. First, we prove that if a Sasakian manifold M admits a nontrivial$\rho $-Einstein soliton $(M,g,V,\lambda )$, then M is $\mathcal {D}$-homothetically fixed null $\eta $-Einstein and the soliton vector field V is Jacobi field along trajectories of the Reeb vector field $\xi $, nonstrict infinitesimal contact transformation and leaves $\varphi $ invariant. Next, we find two sufficient conditions for a compact $\rho $-Einstein almost soliton to be trivial (Einstein) under the assumption that the soliton vector field is an infinitesimal contact transformation or is parallel to the Reeb vector field $\xi $.

Our main result is the $\mathbb {\mathcal {C}}^{0}$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.