In this note we consider two topics involving the relationship between the symplectic capacity and the mean width of convex bodies in ℝ^2n. We first describe an alternative path from the symplectic Brunn-Minkowski inequality of Artstein-Avidan and Ostrover to another inequality, established by the same authors, that relates the capacity and mean width of convex bodies. This new path is less direct but it relates these inequalities to the quermassintegrals of convex bodies and to the local version of Viterbo's conjecture established by Abbondandolo and Benedetti for domains sufficiently close to the ball. We then consider the problem of identifying convex bodies whose mean width cannot be decreased by natural classes of symplectomorphisms. We state a conjectured characterization of convex bodies whose mean width is already minimal among all their symplectic images. To test this conjecture we identify a simple class of quadratic convex bodies whose mean width can not be decreased by linear symplectic maps near the identity. We then identify a subset of these examples that fail to satisfy the toric conditions of the conjecture, and show that one can find a nonlinear symplectomorphism that decreases their mean width.

For a large class of toric domains in ℝ^4 we determine which product Lagrangian tori can be mapped into the domain by a Hamiltonian diffeomorphism. In other words, we compute the Hamiltonian shape invariant of these toric domains, as defined by Hind and Zhang. The argument relies on new intersection results for product Lagrangian tori in symplectic polydisks. For Hamiltonian diffeomorphisms which map certain Lagrangian product tori back into the polydisk, we establish intersections between the images and a one-parameter family of product Lagrangian tori that includes (is based at) the original torus. For symplectic polydisks with area ratios less than two, we strengthen this to establish intersections between the Hamiltonian images and the original Lagrangian torus. As a soft complement to these intersection results we also present an embedding construction which demonstrates that this intersection rigidity vanishes when the one-parameter family of product Lagrangian tori is replaced by a natural packing by Lagrangian tori.

In this paper we settle three basic questions concerning the Gutt-Hutchings capacities. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities together with the volume, do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. We also establish two independence properties. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume. We also prove that the capacities are mutually independent by constructing, for any $j \in \mathbb{N}$, a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the $j^{th}$. The constructions underlying these results are not exotic. They are convex and concave toric domains. A key to the progress made here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains which holds under an additional symmetry assumption. This simplification allows us to identify new blind spots of the capacities which are used to construct the desired examples.

We consider the problem of packing a symplectic manifold with integral Lagrangian tori, that is, Lagrangian tori whose area homomorphisms take only integer values. We prove that the Clifford torus in S-2 x S-2 is a maximal integral packing, in the sense that any other integral Lagrangian torus must intersect it. In the other direction, we show that in any symplectic polydisk P (a, b) with a, b > 2, there is at least one integral Lagrangian torus in the complement of the collection of standard product integral Lagrangian tori.

In this work we investigate subsets of $\mathbb{R}^{2n}$ with the property that their mean width can not be decreased by the action of natural classes of symplectomorphisms. A common theme of our results is that toric symmetry is a preferred feature of subsets that are in optimal symplectic position with respect to the mean width. Among other things, we prove that the mean width of a toric domain can not be decreased by any linear symplectic map. We also show that toric convex domains are critical points of the mean width under all symplectic deformations.

In this work, we study convex bodies in ^2n with the property that their mean width cannot be infinitesimally decreased by symplectomorphisms. The common theme of our results is that toric symmetry is a preferred feature of convex bodies with this property.

We use topological summaries based on Betti curves to characterize the large-scale spatial distribution of simulated dark matter haloes and galaxies. Using the IllustrisTNG and CAMELS-SAM simulations, we show that the topology of the galaxy distribution is significantly different from the topology of the dark matter halo distribution. Further, there are significant differences between the distributions of star-forming and quiescent galaxies. These topological differences are broadly consistent across all simulations, while at the same time there are noticeable differences when comparing between different models. Finally, using the CAMELS-SAM simulations, we show that the topology of the quiescent galaxies in particular depends strongly on the amount of supernova feedback. These results suggest that topological summary statistics could be used to help better understand the processes of galaxy formation and evolution.

In [1], Artstein-Avidan and Ostrover establish a symplectic version of the classical Brunn-Minkowski inequality where the role of the volume is played by the Ekeland-Hofer-Zehnder capacity. Here we prove that this symplectic Brunn-Minkowski inequality fails to hold for all of the higher index symplectic capacities defined by Gutt and Hutchings in [5].

We establish results concerning the existence and nonexistence of regular $J$-holomorphic cylinders between nested pairs of ellipsoids in $\mathbb{R}^4$.

We correct here an error in the proof of Theorem 1.1 of our paper New obstructions to symplectic embeddings, henceforth referred to as [6].

In this paper we establish new restrictions on the symplectic embeddings of basic shapes in symplectic vector spaces. By refining an embedding technique due to Guth, we also show that they are sharp.

We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekelend and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby-Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits can not be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we partially recover, using only symplectic tools, a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere by Franks and Handel.

The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion have been proposed so far, it remains an open problem to devise -type search procedures with provable performance guarantees. The standard approach to the problem, which involves the minimization of an objective function defined using the Frobenius metric, has inherent difficulties: the objective function is not continuous and the solution set is not closed. To address this problem, we consider an optimization procedure that searches for a column (or row) space that is geometrically consistent with the partial observations. The geometric objective function is continuous everywhere and the solution set is the closure of the solution set of the Frobenius metric. We also preclude the existence of local minimizers, and hence establish strong performance guarantees, for special completion scenarios, which do not require matrix incoherence and hold with probability one for arbitrary matrix size.

A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we reprove Franks' theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from all previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorpisms.

The low-rank matrix completion problem can be stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. There exist several low-complexity algorithms for low-rank matrix completion which focus on the minimization of the Frobenius norm of the matrix projection residue. This optimization framework has inherent difficulties: the objective function is not continuous and the solution set is not closed. To address this problem, we propose a geometric objective function to replace the Frobenius norm: the new objective function is continuous everywhere and the solution set is the closure of the solution set of the Frobenius metric. Furthermore, using the geometric objective function and a simple gradient descent procedure, we are able to preclude the existence of local minimizers, and hence establish strong performance guarantees for special completion scenarios, which do not require matrix incoherence or large matrix size.

We describe a new algorithm, termed subspace evo- lution and transfer (SET), for solving consistent low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix and outputs one low-rank matrix con- sistent with the given observations. The completion task is accom- plished by searching for a column space in the Grassmann mani- fold that matches the incomplete observations. The SET algorithm consists of two parts—subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of the column space. Since the gradient descent algorithm is not guaranteed to converge due to the existence of barriers along the search path, we design a new mechanism for detecting barriers and transferring the esti- mated column space across the barriers. This mechanism consti- tutes the core of the transfer step of the algorithm. The SET algo- rithm exhibits excellent empirical performance for a large range of sampling rates.

We describe a new algorithm, termed subspace evolution and transfer (SET), for solving consistent low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix and outputs one low-rank matrix consistent with the given observations. The completion task is accomplished by searching for a column space in the Grassmann manifold that matches the incomplete observations. The SET algorithm consists of two parts-subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of the column space. Since the gradient descent algorithm is not guaranteed to converge due to the existence of barriers along the search path, we design a new mechanism for detecting barriers and transferring the estimated column space across the barriers. This mechanism constitutes the core of the transfer step of the algorithm. The SET algorithm exhibits excellent empirical performance for a large range of sampling rates.

We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural requirements. In the case of Reeb flows, this leads to simple expressions (purely in terms of the mean indices) for the mean Euler characteristics. These are invariants of the underlying contact structure which are capable of distinguishing some contact structures that are homotopic but not diffeomorphic.

In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four.The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flows supported near the Lagrangian, and the length minimizing properties of these flows with respect to the negative Hofer length functional.