Related papers: Floer-type bipersistence modules and rectangle bar…
We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section.…
The persistence barcode is a well-established complete discrete invariant for finitely generated persistence modules [5] [1]. Its definition, however, does not extend to multi- dimensional persistence modules. In this paper, we introduce a…
The purpose of this thesis is to define a "local" version of Ozsv\'{a}th and Szab\'{o}'s Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology…
We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is…
With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which…
We construct a stable infinity category with objects flow categories and morphisms flow bimodules; our construction has many flavors, related to a choice of bordism theory, and we discuss in particular framed bordism and the bordism theory…
We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets,…
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special…
Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…
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…
A persistence module is a functor $f: \mathbf{I} \to \mathsf{E}$, where $\mathbf{I}$ is the poset category of a totally ordered set. This work introduces saecular decomposition: a categorically natural method to decompose $f$ into simple…
We study the inverse problem for persistent homology: For a fixed simplicial complex $K$, we analyse the fiber of the continuous map $\mathrm{PH}$ on the space of filters that assigns to a filter $f: K \to \mathbb R$ the total barcode of…
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an…
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…
A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal…
We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a…
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…