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Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use…
This is a survey article on classical groups (over arbitrary division rings) and their geometries.
We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko's work.
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
In this paper, we present a constructive generalization of metric and uniform spaces by introducing a new class of spaces, called cover spaces. These spaces form a topological concrete category with a full reflective subcategory of complete…
We describe rings over which every right module is almost injective. We give a description of rings over which every simple module is a almost projective.
We call an affine algebraic supergroup quasireductive if its underlying algebraic group is reductive. We obtain some results about the structure and representations of reductive supergroups.
Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In…
We show that the group of type-preserving automorphisms of any irreducible semi-regular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated (abstractly)…
We determine all spherical algebraic subalgebras in any simple Lie algebra of real rank 1.
We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not…
We provide a simple proof that conformally semi-symmetric spacetimes are actually semi-symmetric. We also present a complete refined classification of the semi-symmetric spacetimes.
We give an intrinsic characterization of all subsets of a doubling metric space that can arise as a member of some system of dyadic cubes on the underlying space, as constructed by M. Christ.
Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups.
We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.
In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact…
We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…
\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are…
A description of all subsemigroups of $M_2(\mathbb{C})$ which are given by a countable intersection of constructible sets is given. Furthermore, it is shown that they are intersections of constructible semigroups.
We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Franc\'{e}s showing that it is NP-complete to decide whether a given simplicial…