Related papers: Geometric analysis on manifolds with ends

The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections…

This is a short survey paper, partly meant as a research announcement. Its purpose is to highlight some aspects of the interplay between quantales, inverse semigroups, and groupoids. Many of the results mentioned have not yet been presented…

Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two notions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle ${}^bTX$. The usual…

We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries

Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature…

This paper reviews the most notable works applying machine learning techniques (ML) in the context of geophysics and corresponding subbranches. We showcase both the progress achieved to date as well as the important future directions for…

Quenched and annealed heat kernel estimates are established for Fontes-Isopi-Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to…

We deal with rigidity results for compact gradient Einstein-type manifolds with nonempty boundaries. As a result, we obtain new characterizations for hemispheres and geodesic balls in simply connected space forms. In dimensions three and…

Manifolds have uses throughout and beyond Mathematics and it is not surprising that topologists have expended a huge effort in trying to understand them. In this article we are particularly interested in the question: `when is a manifold…

The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in…

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds…

Functional geometry is a framework using concepts from geometry to understand the invariance of amplitudes in quantum field theory under a large class of field redefinitions, including those involving derivatives. It is inspired by…

This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…

We construct extremal metrics on the total space of certain destabilising test configurations for strictly semistable K\"ahler manifolds. This produces infinitely many new examples of manifolds admitting extremal K\"ahler metrics. It also…

We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…

In this article we define the analytic torsion of finite volume orbifolds $\Gamma \backslash \mathbb{H}^{2n+1}$ and study its asymptotic behavior with respect to certain rays of representations.

In this paper, we demonstrate that not only the heat kernel techniques are useful for computation of the parity anomaly, but also the parity anomaly turns out to be a powerful mean in studying the heat kernel. We show that the gravitational…

We consider some integral-geometric quantities that have recently arisen in harmonic analysis and elsewhere, derive some sharp geometric inequalities relating them, and place them in a wider context.

We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein's inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for…

We compute the p-widths, $\{\omega_p\}$, for the hemisphere with the standard round metric. This provides the first example of a manifold with boundary for which the $p$-widths are known for all $p$.