Related papers: A construction of algebraic surfaces with many rea…
We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a…
We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In…
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…
We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the…
Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…
We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
Suppose $\Omega, A \subseteq \RR\setminus\Set{0}$ are two sets, both of mixed sign, that $\Omega$ is Lebesgue measurable and $A$ is a discrete set. We study the problem of when $A \cdot \Omega$ is a (multiplicative) tiling of the real line,…
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic…
Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the…
It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…
We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the…
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.
We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of…
We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers up to two, one associates an adelic group. We show that this operation commutes with taking intersections if the surface is…
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…