Related papers: An overview of complex ellipsoids
A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of…
We investigate complex structures on the Oeljeklaus-Toma manifolds. The Oeljeklaus-Toma manifolds are defined using complex embeddings of number fields. By replacing these embeddings with their conjugates, one obtains other manifolds that…
This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.
An unfolding of a polyhedron along its edges is called a vertex unfolding if adjacent faces are allowed to be connected at not only an edge but also a vertex. Demaine et al showed that every triangulated polyhedron has a vertex unfolding.…
In this paper we study the fiber body, that is the extension of the notion of fiber polytopes for more general convex bodies. After giving an overview of the properties of the fiber body, we focus on three particular classes of convex…
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a…
The term complexity derives etymologically from the Latin plexus, which means interwoven. Intuitively, this implies that something complex is composed by elements that are difficult to separate. This difficulty arises from the relevant…
Ellipsoids possess several beautiful properties associated with classical potential theory. Some of them are well known, and some have been forgotten. In this article we hope to bring a few of the "lost" pieces of classical mathematics back…
On an elliptic billard, we study a complex reflexion law introduced by A. Glutsyuk in order to state that the locus of the circumcenters of triangular orbits is an ellipse.
We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a…
A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as…
The Illumination Problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball,…
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in…
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as…
Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission critical systems. For such applications, the identification of an appropriate…
An open set in C^n is pseudoconvex if and only if its intersection with every affine subspace of complex dimension two as seen as an open set in C^2 is pseudoconvex.
This paper studies the configuration spaces of linkages whose underlying graph is a single cycle. Assume that the edge lengths are such that there are no configurations in which all the edges lie along a line. The main results are that,…
The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterising ellipsoids of maximal volume contained in convex bodies.
Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This…
We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random…