Related papers: Existence, Regularity, and Properties of Generaliz…

This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The…

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…

We introduce a ``hybrid'' conjecture which is a common generalisation of the Andr\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid…

Locating apparent horizons is not only important for a complete understanding of numerically generated spacetimes, but it may also be a crucial component of the technique for evolving black-hole spacetimes accurately. A scheme proposed by…

We investigate the cosmological implications of generalized mass-to-horizon entropy, a two-parameter extension of the standard Bekenstein entropy based on the mass-to-horizon relation. Assuming the entropy balance relation, we derive the…

We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or…

We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving…

We show that min-max minimal hypersurfaces can be localized. As a consequence, we obtain the sharp generalization to complete manifolds of the famous Almgren-Pitts min-max theorem in closed manifolds. We use this result to prove the…

We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension $11$ in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, $n+1 \geq 12$, we prove in the…

We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…

Formulation of the Penrose inequality becomes ambiguous when the past and future apparent horizons do cross. We test numerically several natural possibilities of stating the inequality in punctured and boosted single- and double- black…

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no…

We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple…

The present talk summarizes the recently progressed state of a systematic re-evaluation of cosmological models that respect the presence of inhomogeneities. Emphasis is given to identifying the basic steps towards an effective (i.e.…

A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are…

We investigate the validity of the generalized second law of thermodynamics on the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk. We examine the evolution of the apparent horizon entropy extracted through relation…

Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are…

In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. Demol [3]. This generalization is useful for solving many practical problems in which more than one constraint are involved.…

We make the case for the existence of a, hitherto unknown and unobserved, hierarchy of ever more compact cosmic objects in the universe. This hypothesis is based on i) the assumption of "elementary" particle sub-constituents on several…

We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are…