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We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally,…

The recent non-calculus proof of Kepler's first law succeeds because of an obscure, but valid property of the ellipse.

The purpose of these short notes is to provide a concise proof of a celebrated theorem by Dembo, Peres, Rosen and Zeitouni, which settles the leading order of cover times in the small-\varepsilon regime.

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…

We survey results on the problem of covering the space ${\mathbb R}^n$, or a convex body in it, by translates of a convex body. Our main goal is to present a diverse set of methods. A theorem of Rogers is a central result, according to…

The main theorem of "S. J. Kov\'acs: The cone of curves of a K3 surface, Math. Ann. 300 (1994), no. 4, 681-691" is proved in arbitrary characteristic. The proof is essentially the same as in the original paper where it was stated only over…

A short proof of the Mazur-Ulam theorem concerning isometries of real normed spaces.

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type…

In this note we give a detailed proof of a theorem of Aubin.

We give a new proof of the existence of designs, which is much shorter and gives better bounds.

Covering space theory is used to construct new examples of buildings.

For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the…

The algorithm of Edelsbrunner for surface reconstruction by ``wrapping'' a set of points in R^3 is described.

We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by M\"obius, using hyperbolic geometry.

We expose here a short proof of Cramer's theorem in R based on convex duality.

We prove a generalization of Lopes's theorem, that is, of the converse of Brolin's theorem.

We provide a simple proof of Kamp's theorem.

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.

In this paper we prove a desingularization theorem for Legendrian surfaces that are the conormal of a quasi-ordinary hypersurface.

In this note, we present a simple directed graph proof of Sharkovsky's theorem.