Related papers: Metrics on diagram groups and uniform embeddings i…
We consider low-distortion embeddings for subspaces under \emph{entrywise nonlinear transformations}. In particular we seek embeddings that preserve the norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where $Z$ is a…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a *-structure, conjugate-linear on the…
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space,…
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically use…
Human mathematics (HM), the mathematics humans discover and value, is a vanishingly small subset of formal mathematics (FM), the totality of all valid deductions. We argue that HM is distinguished by its compressibility through…
In this work, we stress the existence of isomorphisms which map complex contours from the upper half to contours in the lower half of the complex plane. The metric operator is found to depend on the chosen contour but the maps connecting…
We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,\mu)$ is…
The holographic entropy bound is used to estimate the quantum-gravitational discreteness of inflationary perturbations. In the context of scalar inflaton perturbations produced during standard slow-roll inflation, but assuming that…
A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of…
By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which…
A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of…
Motivated by Wheeler's bottom up pregeometry, we introduce a pregeometric approach that does not assume Wheeler's probability amplitudes for establishing spacetime neighborhoods. Rather, a non-trivial metric is produced via the concept of a…
We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we…
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…
To a finite group G one can associate a tower of wreath products S_n[G]. It is well known that the graded direct sum of the Grothendieck groups of the categories of finite dimensional complex representations of these groups can be given the…
The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor…
We study the cusped Hitchin component consisting of (conjugacy classes of) cusped Hitchin representations of a torsion-free geometrically finite Fuchsian group into PSL(d,R). We produce pressure metrics associated to the first fundamental…
Dowdall and Taylor observed that given a finite-index subgroup of a free group, taking covers induces an embedding from the Outer Space of the free group to the Outer Space of the subgroup, that this embedding is an isometry with respect to…
We study properties of twisted unions of metric spaces introduced by Johnson, Lindenstrauss, and Schechtman, and by Naor and Rabani. In particular, we prove that under certain natural mild assumptions twisted unions of $L_1$-embeddable…