The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proved by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg's proof of the Conley Conjecture for closed symplectically aspherical manifolds.Comment: 14 pages, 1 figure, version 2: some corrected typos and added references, one added remark on possible generalizatio

Cuplength estimates in Morse cohomology

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms, and solutions to a Dirac-type equation.Comment: 25 pages, 1 figure, appeared online in Journal of Topology and Analysi

Hyperk\"ahler Arnold Conjecture and its Generalizations

We generalize and refine the hyperk\"ahler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperk\"ahler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley-Zehnder proof of the Arnold conjecture for the torus.Comment: 13 page

Identification of 12 new susceptibility loci for different histotypes of epithelial ovarian cancer.

To identify common alleles associated with different histotypes of epithelial ovarian cancer (EOC), we pooled data from multiple genome-wide genotyping projects totaling 25,509 EOC cases and 40,941 controls. We identified nine new susceptibility loci for different EOC histotypes: six for serous EOC histotypes (3q28, 4q32.3, 8q21.11, 10q24.33, 18q11.2 and 22q12.1), two for mucinous EOC (3q22.3 and 9q31.1) and one for endometrioid EOC (5q12.3). We then performed meta-analysis on the results for high-grade serous ovarian cancer with the results from analysis of 31,448 BRCA1 and BRCA2 mutation carriers, including 3,887 mutation carriers with EOC. This identified three additional susceptibility loci at 2q13, 8q24.1 and 12q24.31. Integrated analyses of genes and regulatory biofeatures at each locus predicted candidate susceptibility genes, including OBFC1, a new candidate susceptibility gene for low-grade and borderline serous EOC

The Role of Imaging in Measuring Disease Progression and Assessing Novel Therapies in Aortic Stenosis

Aortic stenosis represents a growing health care burden in high-income countries. Currently, the only definitive treatment is surgical or transcatheter valve intervention at the end stages of disease. As the understanding of the underlying pathophysiology evolves, many promising therapies are being investigated. These seek to both slow disease progression in the valve and delay the transition from hypertrophy to heart failure in the myocardium, with the ultimate aim of avoiding the need for valve replacement in the elderly patients afflicted by this condition. Noninvasive imaging has played a pivotal role in enhancing our understanding of the complex pathophysiology underlying aortic stenosis, as well as disease progression in both the valve and myocardium. In this review, the authors discuss the means by which contemporary imaging may be used to assess disease progression and how these approaches may be utilized, both in clinical practice and research trials exploring the clinical efficacy of novel therapies

Variation on the Theme of the Conley Conjecture

We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes on the second fundamental group. In particular, this removes the rationality condition from similar theorems by Ginzburg and Gürel. The proof in the irrational case involves several new ingredients including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. For this proof, we develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition. One of the summands only depends on the behavior of the Hamiltonian in a fixed open set and enables us to use tools from more restrictive cases in the proof of the Conley conjecture. We also prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians, which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits

PERIODIC REEB ORBITS ON PREQUANTIZATION BUNDLES

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold M, pinched between two circle bundles whose ratio of radii is less than root 2 carries either one short simple periodic orbit or carries at least cuplength(M) + 1 simple periodic Reeb orbits