Related papers: Bounds on spectral norms and barcodes

We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the…

This paper studies how symplectic invariants created from Hamiltonian Floer theory change under the perturbations of symplectic structures, not necessarily in the same cohomology class. These symplectic invariants include spectral…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or…

The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing…

Let $N$ be a closed manifold and $U \subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds that are…

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame…

In this paper, we show that the pointwise finite-dimensional two-parameter persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of interlevel filtered Floer homology, is rectangle-decomposable. This allows for the…

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper "Symplectic topology as the geometry of generating functions," they have been defined in various contexts, mainly via…

This is a mixture of survey article and research anouncement. We discuss Instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian…

We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a ``large enough" tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an…

We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this…

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of…

In this article, we modify the classical Floer complex $CF(L_0,L_1)$ of a pair of two compact exact Lagrangian submanifolds $L_0,L_1$ of an exact symplectic 2-manifold $M$ into a $\mathbb{Z}_2[T]$-complex $CF_h(L_0,L_1)$, whose differential…

The purpose of this paper is to extend the construction of the PSS-type isomorphism between the Floer homology and the quantum homology of a monotone Lagrangian submanifold $L$ of a symplectic manifold $M$, provided that the minimal Maslov…

We develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves…

We review our recent results on the on-shell description of sine-Gordon model with integrable boundary conditions. We determined the spectrum of boundary states by closing the boundary bootstrap and gave a derivation of Al.B.…

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the…

Motivated by the ergodic closing lemma of Ma\~n\'e, we investigate the $C^\infty$ closing lemma in higher-dimensional Hamiltonian systems, with a focus on the statistical behavior of periodic orbits generated by $C^\infty$-small…

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric…