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Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from the shape by applying a multiscale approach to the calculus of the fractal dimension of that shape. The fractal…
The purpose of this paper is to provide a both comprehensive and summarizing account on recent results about analysis and geometry on configuration spaces $\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to a complete…
Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…
Riemannian geometry is a mathematical field which has been the cornerstone of revolutionary scientific discoveries such as the theory of general relativity. Despite early uses in robot design and recent applications for exploiting data with…
The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space…
We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems…
Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article…
Work in progress concerning alternative formalizations of arithmetic.
In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological…
The root is an important organ of a plant since it is responsible for water and nutrient uptake. Analyzing and modelling variabilities in the geometry and topology of roots can help in assessing the plant's health, understanding its growth…
The article is devoted to the investigation of properties of quasi-invariant measures with values in non-Archimedean fields such as: convolutions of measures and functions; continuity of functions of measures; non-associative noncommutative…
Aortic shape analysis plays a key role in cardiovascular diagnostics, treatment planning, and understanding disease progression. We present a robust, fully automated pipeline for aortic shape analysis from cardiac MRI, combining deep…
A suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: --1)…
An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with `Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths…
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
Bargmann invariants, a class of gauge-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the geometric structure of quantum mechanics. This survey offers a…
Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…
Topological data analysis provides a collection of tools to encapsulate and summarize the shape of data. Currently it is mainly restricted to \emph{mapper algorithm} and \emph{persistent homology}. In this paper we introduce new…