Related papers: A Guided Tour to Normalized Volume

The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.

We give a negative answer to a question by J.M. Landsberg on the nature of normalizations of orbit closures. A counterexample originates from the study of complex, ternary, cubic forms.

This article provides a concise overview of recent theoretical results concerning the theory of vortons, which are defined to be (centrifugally supported) equilibrium configurations of (current carrying) cosmic string loops. Following a…

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.

In this document, we make a round up of the theory of asymptotic normality of sums of associated random variables, in a coherent approach in view of further contributions for new researchers in the field. (Version 01)

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…

Using a particular Hilbert space representation of minimum-length deformed quantum mechanics, we show that the resolution of the wave-function singularities for strongly attractive potentials, as well as cosmological singularity in the…

Based on a study of recently proposed solution of 2 dim. black hole we argue that the space-time singularities of general relativity may be described by topological field theories (TFTs). We also argue that in general TFT is a field theory…

A nonlocal generalization of quantum field theory in which momentum space is the space of continuous maps of a circle into $\mathbf{R}^4$ is proposed. Functional integrals in this theory are proved to exist. Renormalized quantum field model…

The aim of the paper is to start to develop the most general theory of localizations/inversion. Several new concepts are introduced and studied.

This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind…

In this paper we investigate the vanishing of cosmological singularities by quantization. Starting from a 5d Kaluza--Klein approach we quantize, as a first step, the non--spherical metric part and the dilaton field. These fields which are…

The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.

We show that diagonalization, products and lower truncations preserve the property of being a denormalized volume polynomial. We also discuss an application to poset inequalities.

We explore an asymptotic behavior of entropies for sums of independent random variables that are convolved with a small continuous noise.

We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.

This article presents a complete second order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results don't require the existence of a radius of stabilisation. Hence they can be applied to…

After quick survey of some key results and open questions about the structure of singularities of minimal surfaces, we discuss recent work~\cite{Sim23} on singularities of stable minimal hypersurfaces, including some simplifications of the…

We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for…

A brief overview is presented of recent developments concerning resummed small-x evolution, based upon the renormalization group equation. The non-singlet and singlet structure functions are discussed for both polarized and unpolarized…