ALT We introduce a notion of size for an overtwisted disk, defined in terms of the Shelukhin-Hofer energy required to stabilise Legendrians in its vicinity. We compute this size for the standard 3-dimensional overtwisted disk, which we use to prove that certain explicit product hyperdisks in higher-dimensional contact manifolds are overtwisted (and their overtwisted size can be computed explicitly as well). We then use the overtwisted size to establish a number of quantitative results on contact flexibility. Concretely, we (1) obtain quantitative h–principles for Legendrians in the complement of an overtwisted disk, (2) construct overtwisted-at-infinity contact manifolds whose Legendrian Shelukhin-Hofer metric vanishes, (3) estimate overtwisted size under contact products and under products with Liouville manifolds, and (4) derive action bounds for contractible Reeb orbits in closed overtwisted contact manifolds.

ALT For every integer n≥5, we construct a closed aspherical symplectic manifold of real dimension 2n with positive first Chern degree. Each example is a flat torus bundle over the same compact hyperbolic surface, with arithmetic monodromy arising from an explicit indefinite quaternion division algebra over ℚ. We identify its symplectic first Chern class with the pullback of the canonical class of the base and compute its degree explicitly. Asphericity rules out nonconstant pseudoholomorphic spheres for every tame almost complex structure. Thus these examples give a negative answer to the symplectic uniruledness question of Boucksom–Demailly–Păun–Peternell in every even real dimension at least ten.

ALT We construct several families of torus equivariant Poisson structures on moment-angle manifolds and their quotients such as quasitoric manifolds. We study the symplectic foliation and Poisson cohomology groups associated with these structures.

ALT For a Hermitian line bundle E→ Y over a Riemannian manifold (Y^n,g), the self-dual Abelian–Higgs energy is defined by \En^Y_\eps,g(u,a)=frac12π\int_Y (|∇_a u|g²+\eps²|F_a|g²+ (1-|u|²)²/4\eps²)dVg, where u is a smooth section of E, a is a unitary connection, and F_a is its curvature. Pigati and Stern studied the energy as a codimension-two analogue of the Allen–Cahn energy to study minimal submanifolds in a Riemannian manifold. In this paper, we use this functional to define a nondecreasing sequence of symplectic capacities {c_k^AH}_k≥1 in dimension four by a min–max construction. In particular, we prove that c₁^AH is normalized on the ball and the cylinder. For the upper bound on the cylinder, we combine Gromov's theory of families of J-holomorphic spheres with Taubes's planar vortex. For the lower bound on the ball, we use the properties of the Abelian–Higgs theory developed by Pigati–Stern, together with a geometric estimate on smooth approximations to the double of the ball and an e

ALT We construct an elliptic analogue of the Alekseev-Meinrenken path fibration for a simply connected complex Lie group with a nondegenerate invariant symmetric bilinear form. A natural 2-form on the correspondence space induces a holomorphic symplectic groupoid over the space of (0,1)-connections and a holomorphic quasi-symplectic groupoid over the holomorphic loop group. The correspondence space defines a principal holomorphic Morita bibundle with compatible differential forms.

ALT We settle several questions about Hofer's geometry of the two-sphere. We prove that the space of equators has infinite Hofer diameter, construct simple commutators of arbitrarily large Hofer norm, and exhibit a Hamiltonian diffeomorphism whose iterates escape to infinity in Hofer distance from the set of autonomous diffeomorphisms. Finally, we construct a bi-Lipschitz group monomorphism from the free group of rank two, equipped with its conjugation-invariant generator norm, into the Hamiltonian group equipped with Hofer's metric. The proof rests on an interplay between Lagrangian Floer theory in the fourth symmetric product of the sphere and combinatorial group theory. This paper was written with substantial assistance from AI.

ALT We study a construction for producing Finsler metrics on the sphere from convex bodies in ℝ⁴ using the quaternions. Among several applications, we prove the weak Viterbo conjecture for convex bodies in ℝ⁴ that are invariant under the natural action of the quaternion group Q. This is the first instance where Viterbo's systolic inequality is established for convex bodies with a finite group symmetry, addressing a question of Haim-Kislev-Ostrover. Moreover, a smooth Q-symmetric convex domain that achieves equality is symplectomorphic to the ball, without any assumption of strict or uniform convexity. Finally, we show that any link of minimal Reeb orbits on the boundary of a uniformly convex Q-symmetric convex body is transversely isotopic to a union of Hopf fibers, strengthening a result of Abbondandolo-Edtmair-Kang in the Q-symmetric setting.

ALT The attention mechanism underpins modern large language models, yet their dynamics remain only partially understood. Recent perspectives interpret attention blocks as interacting particle systems, whose mean-field limits correspond to gradient flows of energy functionals on spaces of probability densities on ℝᵈ equipped with Wasserstein-type metrics. We extend this viewpoint by deriving accelerated attention blocks from inertial, Nesterov-type damped Hamiltonian dynamics on these density spaces, developing the geometry and differential calculus needed to make the construction rigorous. In particular, we introduce a local-mobility geometry for the Otto-Wasserstein and softmax-Wasserstein settings, and a quotient Stein geometry that resolves the degeneracy of the linear-attention kernel to yield a well-defined metric and differential calculus. We provide rigorous foundations when softmax attention admits a gradient-flow formulation and when its inertial counterpart admits a damped Hamilt

ALT The documents [OAI26a, OAI26b, OAI26c] claimed to prove cases of the Hodge conjecture via homological mirror symmetry, before being withdrawn by OpenAI. We discuss the source of the error (but do not analyse the rest of the argument).

ALT Every singular subalgebroid, and in particular every singular foliation G, determines a universal Poisson C∞-scheme Spec(ℱ(G)), which is the analogue of the linear Poisson structure on the dual of a Lie algebroid. We show that the underlying space Xℱ(G) of Spec(ℱ(G)) decomposes as a disjoint union of symplectic manifolds. Furthermore, we extend the Helffer-Nourrigat cone to a singular subalgebroid and show that it is an embedded Poisson C∞-subscheme of Spec(ℱ(G)).

ALT We prove multiplicity results for geometrically distinct brake Reeb orbits on real prequantization S¹-bundles under geometric and dynamical assumptions. The lower bounds are expressed in terms of the sum of the mod-two Betti numbers of the real locus of the base. Our proofs combine Lagrangian Rabinowitz Floer homology with the index iteration theory for brake orbits. As an application, we obtain a sharp lower bound for the number of brake geodesics on Finsler spheres with pinched flag curvature.

ALT We present a quantitative exploration of the behaviour of displacement energy of Lagrangians under stabilization. In particular, we completely characterize the displacement behaviour of toric fibres in Fano 4-manifolds. Our approach relates the geometry of the moment polytope's caustic to the structure of the (bulk-deformed) superpotential and to the presence of symmetric probes.

ALT We introduce the notion of a symplectophagy, which is a symplectic isotopy satisfying conditions reminiscent of the symplectic camel. We present new symplectic rigidity results which take the form of obstructions to the existence of symplectophagies in certain situations. The main tool is the relative symplectic cohomology recently introduced by Varolgunes, as well as its extension to pairs, due to the authors and L. Polterovich, for which we establish an invariance property, which is the chief technical novelty of the present paper. An important role is also played by the recent calculations of relative symplectic cohomology in complex projective spaces due to Dickstein-Ganor.

ALT In this paper we study the multiplicity of contractible closed Reeb orbits on prequantization circle bundles. Building on the existing multiplicity result Theorem 1.3 for closed Reeb orbits on prequantization S¹-bundles, we weaken the index assumptions by requiring only that the supporting contact form has no contractible periodic Reeb orbits with mean Conley-Zehnder index zero and no contractible good periodic Reeb orbits with Conley-Zehnder index 0, and we obtain a lower bound for the number of geometrically distinct contractible closed Reeb orbits carried by contact forms α on a prequantization S¹-bundle under certain natural topological assumptions (Theorem 1.6). In dimension five we further relax the index assumptions and show that, under a mild index condition imposed only on simple contractible periodic orbits and the same topological assumptions on the prequantization S¹-bundle, every non-degenerate supporting contact form carries at least dim H*(B;ℚ) geometrically distinct con

ALT We prove that four periodic curves in ℂ inscribe a family of isosceles trapezoids for every angle θ∈(0,π), whose similarity types are determined by the signed areas between the quotients of the curves in the infinite cylinder. In the smooth case, the problem is identified with a Lagrangian intersection problem in ℝ× S¹×ℂ, while the continuous case follows by a convergence argument.

ALT We give combinatorial criteria for the existence of left-invariant symplectic forms on the positive and full path Lie groups of finite acyclic quivers. For the nilpotent positive path Lie groups, the criterion also determines whether their compact quotients admit symplectic forms and forces the nilpotency step to be at most three. For the completely solvable full path Lie groups, we give a direct combinatorial criterion and an equivalent recursive criterion. In the symplectic case, the recursive procedure is realized by successive Hamiltonian reductions ending at an abelian symplectic Lie group. We characterize the existence of Lagrangian normal subgroups for a given form in both families, with applications to Lagrangian torus fibrations of compact nilmanifolds. We also describe the cohomology rings of the compact symplectic nilmanifolds and determine all Lefschetz ranks.

ALT We prove a criterion for detecting exotic symplectomorphisms of strong fillings of closed contact manifolds. The criterion concerns symplectomorphisms induced by free contact circle actions on the boundary, and it is expressed purely in topological terms. As an application, we show that one-point blow-ups of an infinite family of Brieskorn varieties admit exotic compactly supported symplectomorphisms.

ALT We prove the unboundedness of the spectral distance on the space of Lagrangians Hamiltonian isotopic to the zero section within the disc cotangent bundle of a closed manifold satisfying some topological assumptions. This disproves the generalization of a conjecture due to Claude Viterbo. The proof relies on a direct computation of the spectral distance between the zero section and its image by a magnetic Hamiltonian diffeomorphism via generating functions. We also classify all surfaces for which the conjecture holds over ℤ₂ via a Floer-theoretic argument for covering spaces.

ALT We resolve an open problem posed by Tao related to the Toeplitz square peg conjecture. We prove that four simple closed polygonal curves in the cylinder, each winding once in the positive direction, jointly inscribe a square whenever the alternating sum of their ∫ y dx integrals vanishes. The proof reduces the problem to the non-displaceability of a pair of Lagrangian tori, which we establish using Lagrangian Floer homology.

ALT In this paper, we study the role of local symplectic groupoid multiplication in the construction of approximation methods for Hamiltonian flows on the underlying Poisson manifold. We show it can be used to construct a novel multiplicative type of approximating sequences for the relevant Lagrangian bisection. We describe the corresponding theoretical results, their practical implementation into approximation methods on coordinate charts via generating functions, and illustrative examples.