Related papers: Autonomous Hamiltonian flows, Hofer's geometry and…

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian…

We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.

We show that, given a complete Liouville manifold, any homogeneous quasi-morphism on its Hamiltonian group, which satisfies a strengthened version of Hofer continuity called stability, must vanish. This partially addresses a conjecture due…

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by…

We will study homological stability of the diffeomorphism groups of the manifolds $W_{g,1}:=D^{2n} \# (S^n \times S^n)^{\#g }$ using $E_k$-algebras. This will lead to new improvements in the stability results, especially when working with…

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean…

We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations…

While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten…

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about…

This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description…

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g,$ and let $\beta$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^*M$ that…

We show that the diffeomorphisms of an extended phase space with time, energy, momentum and position degrees of freedom that leave invariant the symplectic 2-form and and a degenerate orthogonal metric dt^2 locally satisfy Hamilton's…

The Hamiltonian operator describing a quantum particle on a path often extends holomorphically to a complex neighborhood of the path. When it does, it can be seen as the local expression of a complex projective structure, and its…

We reformulate the persistent (co)homology of simplicial filtrations, viewed from a more algebraic setting, namely as the (co)homology of a chain complex of graded modules over polynomial ring $K[t]$. We also define persistent (co)homology…

Two-dimensional free-surface flow over localised topography is examined with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a…

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the…

We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,\omega) which upon passing to homology yields ring isomorphisms with the big quantum…

In this article we show that the braid type of a set of $1$-periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric $d_{\rm…

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same…