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Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the…

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are…

If $\Sigma$ and $\Sigma'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense…

Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance d_B between their PDs. A potential drawback of this pipeline is that point clouds sampled from…

We prove that persistence diagrams with the p-Wasserstein distance form the universal p-subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, barcodes, and to…

Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to show that any tangential homotopy can be approximated by a homotopy of topological embeddings with mild singularities. This concept plays an important role in…

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for…

Persistent homology maps a simplicial complex filtered by elements in $\mathbb R$ to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \infty \} \mid b < d \}$ called (finite) persistence diagrams.…

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are…

We introduce a number of tools for finding and studying \emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichm\"{u}ller space with either the Teichm\"{u}ller or…

In this paper, we generalize the result from L. Polterovich and E. Shelukhin's paper stating that Hofer distance from time-dependent Hamiltonian diffeomorphism to the set of p-th power Hamiltonian diffeomorphism can be arbitrarily large to…

Let $P$ be a poset. We define a new homotopy theory of suitably nice $P$-stratified topological spaces with equivalences on strata and links inverted. We show that the exit-path construction of MacPherson, Treumann, and Lurie defines an…

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in…

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…

This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…

We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special…

The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…

The purpose of this paper is to make a comprehensive connection between the basic results and properties derived from the two kinds of topologies (namely the $(\epsilon,\lambda)-$topology introduced by the author and the stronger locally…

What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic…

Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene…