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Formal orbifolds are defined in higher dimension. Their \'etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to…

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the…

Survey about extremum of eigenvalues of geometric operators within a conformal class of a compact riemannian manifold.

In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the…

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to…

Any subanalytic germ $(X,0) \subset (\mathbb R^n,0)$ is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest…

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and…

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is…

A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of…

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with…

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…

We define and study a natural category of graph limits. The objects are pairs $(\pi,\mu)$, where $\pi$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $\mu$ (the…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed…

In this paper, Lusternik-Schinrelmann and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective…

We provide a class of geometric convex domains on which the Carath\'eodory-Reiffen metric, the Bergman metric, the complete K\"ahler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other.…

The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of experiments in applied statistics. In this paper we study some…

We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the derivative ratio approximates, on arbitrary…

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants…

This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors…

We consider a problem on maximizing the height of vertical flight of a material point ("meteorological rocket") in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The…