Related papers: Non-existence of orthogonal complex structures on …

We show the non-existence of sections of arithmetic fundamental groups of open p-adic annuli of small radii, this implies the non-existence of sections of arithmetic fundamental groups of formal boundaries of formal germs of p-adic curves.

All finite simple self $2$-distance graphs with no $4$-cycle, diamond, or triangles with a common vertex are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graphs.

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations.…

We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities…

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; there exist stable…

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be…

The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…

Given an $n$-dimensional Riemannian sphere conformal to the round one and $\delta$-pinched, we show that it does not contain any closed stable minimal submanifold of dimension $2\le k\le n-\delta^{-1}$.

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced $\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian geometry and string theory, since…

We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution…

Complete sets of mutually unbiased bases are only known to exist in prime-power dimensions. We will describe a few approaches to the problem proving the (non)-existence of four mutually unbiased bases in dimension 6. These will include the…

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…

We remark on two different free-boundary problems for holomorphic curves in nearly-K\"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally must be…

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

We show that there are infinitely many nonisomorphic quandle structures on any topogical space $X$ of positive dimension. In particular, we disprove the conjecture, asserting that there are no nontrivial quandle structures on the closed…

We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The…

Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we…

We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitray small neighborhood of the curve, any…

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.