Related papers: Projective Rectangles: Harmonic Conjugation

In this study, the properties of convex pentagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the rotationally symmetric tilings are formed by concave octagons that are generated by two convex pentagons…

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two,…

A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak…

In a space-time, a conformal structure is defined by the distribution of light-cones. Geodesics are traced by freely falling particles, and the collection of all unparameterized geodesics determines the projective structure of the…

We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal…

This paper is devoted to the study of conformal and projective structures, and especially their connections, in the language of 2-frames, or $G$-structures of 2nd-order. While their normal Cartan connections are well-known, we use the…

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle…

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical…

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their…

We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…

The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree…

A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…

A harmonic mapping is a univalent harmonic function of one complex variable. We define a family of harmonic mappings on the unit disk whose images are rotationally symmetric rosettes with $n$ cusps or n nodes, where $n \ge 3$. These…

We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions…

Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric…

We study the realization spaces of matroids and hyperplane arrangements. First, we define the notion of naive dimension for the realization space of matroids and compare it with the expected dimension and the algebraic dimension, exploring…

One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…

Let $s,n \ge 2$ be integers. We give a qualitative structural description of every matroid $M$ that is spanned by a frame matroid of a complete graph and has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that every…