arXiv research

Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler…

We discuss closed symplectic 4-manifolds which admit full symplectic packings by $N$ equal balls for large $N$'s. We give a homological criterion for recognizing such manifolds. As a corollary we prove that ${\Bbb C}P^2$ can be fully packed by $N$ equal balls for every $N\geq 9$.

Researchers show symplectic packing by ellipsoids is unobstructed for various manifolds.

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…

Researchers describe ECC of concave lens spaces and compute symplectic capacities.

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

We prove that the space of symplectic packings of ${\Bbb C}P^2$ by $k$ equal balls is connected for $3\leq k\leq 6$. The proof is based on Gromov-Witten invariants and on the inflation technique due to Lalonde and McDuff.

The paper solves symplectic embedding problems in higher dimensions, proving new embedding conditions.

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacitie…

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In partic…

Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.

Proves rigidity of circle packings in the plane, generalizing previous work.

The paper studies circle packings using renormalization and subdivision rules.

The paper studies rigidity of sphere packings on 3D manifolds with boundary.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

Study generates infinite circle packings with a specific property.

Paper proves circle packings converge to Riemann mapping for Jordan domains.

Proves rigidity of sphere packings on 3D manifolds.

Paper introduces new flows to find circle packings with specific curvature.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

The paper solves the existence problem of sphere packings in higher dimensions.

Projective rigidity of circle packings on complex surfaces proved.

The paper studies rigid sphere packings on 3D manifolds with boundary.

Study on non-orientable surfaces for maximal disc packings.

Grassmannian packings improve CNN kernels' diversity and reduce sparsity.

This paper optimizes neural network training by packing multiple models on a single GPU.

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Study of rod packings in 3-torus using 3-manifold geometry.

Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.

Paper proves rigidity of inversive distance circle packings.

Minimal surfaces created from tiny circle packings changes.

Paper constructs hyperbolic metrics using circle packings and curvature parameters.

Paper calculates ball number of links using Lorentz geometry and circle packing.

Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.

Study on packing links with geometric constraints.

New theorem proves rigidity of circle packings in hyperbolic geometry.

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.

The paper connects Apollonian packings to knot theory and improves link representations.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

The paper solves circle packings on surfaces with boundaries.

The paper deforms circle packings on surfaces with constant curvature.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

Circle packings on compact surfaces simplified.

Properness proven for circle packings and Delaunay patterns on complex projective structures.

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…

The study finds upper bounds on disk packings on hyperbolic surfaces.