Dan Cristofaro‐Gardiner
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View article: Boundaries of open symplectic manifolds and the failure of packing stability
Boundaries of open symplectic manifolds and the failure of packing stability Open
A finite volume symplectic manifold is said to have “packing stability” if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it h…
View article: Curvy points, the perimeter, and the complexity of convex toric domains
Curvy points, the perimeter, and the complexity of convex toric domains Open
We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems; a natural setting for this is a generalized version of convex toric domain which we also study, where ther…
View article: Low-action holomorphic curves and invariant sets
Low-action holomorphic curves and invariant sets Open
We prove a compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy. In the limit, we extract a family …
View article: On the large-scale geometry of domains in an exact symplectic 4-manifold
On the large-scale geometry of domains in an exact symplectic 4-manifold Open
We show that the space of open subsets of any complete and exact symplectic $4$-manifold has infinite dimension with respect to the symplectic Banach-Mazur distance; the quasi-flats we construct take values in the set of dynamically convex…
View article: Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture
Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture Open
We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjec…
View article: On the agreement of symplectic capacities in high dimension
On the agreement of symplectic capacities in high dimension Open
A theorem of Gutt-Hutchings-Ramos asserts that all normalized symplectic capacities give the same value for monotone four-dimensional toric domains. We generalize this theorem to arbitrary dimension. The new ingredient in our proof is the …
View article: Boundaries of open symplectic manifolds and the failure of packing stability
Boundaries of open symplectic manifolds and the failure of packing stability Open
A finite volume symplectic manifold is said to have "packing stability" if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it h…
View article: Special eccentricities of rational four-dimensional ellipsoids
Special eccentricities of rational four-dimensional ellipsoids Open
A striking result of McDuff and Schlenk asserts that in determining when a\nfour-dimensional symplectic ellipsoid can be symplectically embedded into a\nfour-dimensional symplectic ball, the answer is governed by an "infinite\nstaircase" d…
View article: Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces
Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces Open
This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymp…
View article: A note on the existence of U-cyclic elements in periodic Floer homology
A note on the existence of U-cyclic elements in periodic Floer homology Open
Edtmair-Hutchings have recently defined, using periodic Floer homology, a U-cycle property for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the…
View article: Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms Open
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi inv…
View article: The smooth closing lemma for area-preserving surface diffeomorphisms.
The smooth closing lemma for area-preserving surface diffeomorphisms. Open
Spectral invariants arising from twisted periodic Floer homology have recently played a key role in resolving various open problems in two-dimensional dynamics. We resolve in great generality a conjecture of Hutchings regarding the relatio…
View article: Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids
Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids Open
The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to a…
View article: Contact three-manifolds with exactly two simple Reeb orbits
Contact three-manifolds with exactly two simple Reeb orbits Open
It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegen…
View article: On infinite staircases in toric symplectic four-manifolds
On infinite staircases in toric symplectic four-manifolds Open
An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corne…
View article: Infinite staircases and reflexive polygons
Infinite staircases and reflexive polygons Open
We explore the question of when an infinite staircase describes part of the ellipsoid embedding function of a convex toric domain. For rational convex toric domains in four dimensions, we conjecture a complete answer to this question, in t…
View article: Ehrhart functions and symplectic embeddings of ellipsoids
Ehrhart functions and symplectic embeddings of ellipsoids Open
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that part of the answer is given by an infinite "Fibonacci staircase." Similarly, Frenkel and M\"uller determined when a …
View article: Proof of the simplicity conjecture
Proof of the simplicity conjecture Open
In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked if the same statement holds in dimension $2$. We show that the group of compactly supp…
View article: New examples of period collapse
New examples of period collapse Open
"Period collapse" refers to any situation where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. We study several interesting situations where this occurs, primarily involving triangles. For e…
View article: Symplectic embeddings of products
Symplectic embeddings of products Open
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by …