Related papers: Unexpected hypersurfaces and their consequences: A…
We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent the negation of…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…
The area of research called \textquotedblleft Lineability\textquotedblright% \ looks for linear structures inside exotic subsets of vector spaces. In the last decade lineability/spaceability has been investigated in rather general settings;…
We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of…
A surface embedded in space, in such a way that each point has a neighborhood within which the surface is a terrain, projects to an immersed surface in the plane, the boundary of which is a self-intersecting curve. Under what circumstances…
In this paper, we extend the method developed in [17, 18] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in…
"Most" hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by…
This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.
We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an…
Let $W\subset \mathbb {P}^n$, $n\ge 3$, be a degree $k$ hypersurface. Consider a "general" reducible, but connected, curve $Y\subset \mathbb {P}^n$, for instance a sufficiently general connected and nodal union of lines with $p_a(Y)=0$,…
Unexpectedness is a central concept in Simplicity Theory, a theory of cognition relating various inferential processes to the computation of Kolmogorov complexities, rather than probabilities. Its predictive power has been confirmed by…
We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection…
A survey of special curves, special subvarieties of $\mathcal{M}_g$, and related topics. A large portion of the text discusses various possible interpretation of the word 'special' in this context by giving also concrete examples. One…
We prove power series expansions for the expectations of the number of vertices and missed area of random $L$-convex polygons in planar convex bodies with sufficiently smooth boundaries. Random $L$-convex polygons arise as the intersection…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…
Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane…
This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an…
We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian…
We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$,…