Related papers: Remarks on Lagrangian intersections in toric manif…
We observe that numerous symplectic resolutions can be expressed as intersections of twisted cotangent bundles. Additionally, their dual symplectic resolutions can be derived from intersections of dual twisted cotangent bundles. We…
We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real…
A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal…
We examine the $L^2$-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform…
In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic…
In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we…
Hidden convexity is a powerful idea in optimization: under the right transformations, nonconvex problems that are seemingly intractable can be solved efficiently using convex optimization. We introduce the notion of a Lagrangian dual…
An interesting question in symplectic topology, which was posed by C. H. Taubes, concerns the topology of closed (i.e. compact and without boundary) connected oriented three dimensional manifolds whose product with a circle admits a…
We present a summary of some results from our article [BZ1] and other recent results on the so-called LVMB manifolds. We emphasize some features by taking a different point of view. We present a simple variant of the Delzant construction,…
We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient…
In this paper, we extend the Atiyah--Guillemin--Sternberg convexity theorem and Delzant's classification of symplectic toric manifolds to presymplectic manifolds. We also define and study the Morita equivalence of presymplectic toric…
In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a…
We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral…
In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the $2d$-dimensional annulus and prove a rigidity result for completely integrable ones. More specifically, we consider $1$-parameter families…
Given a smooth Tonelli Hamiltonian on the torus $\mathbb{T}^{n}$ and a $C^{2}$ Lagrangian graph $W \subset T^{*}\mathbb{T}^{n}$ that is invariant under the Hamiltonian flow and contained within a Ma\~n\'e supercritical energy level, we…
By carrying out refined point-wise estimates for the mean curvature, we prove better rigidity theorems of Lagrangian and symplectic translating solitons.
We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series…
The goal of this paper is to classify symplectic toric stratified spaces with isolated singularities. This extends a result of Burns, Guillemin, and Lerman which carries out this classification in the compact connected case. In making this…
We report the results of a theoretical investigation of the stability of a toroidal vortex bound by an interface. Two distinct instability mechanisms are identified that rely on, respectively, surface tension and fluid inertia, either of…