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Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…
We construct a class of representations of the quadratic R-matrix algebra, given by the reflection equation with the spectral parameter, in terms of certain ordinary difference operators. These operators turn out to act as parameter…
We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the…
Already Hermann Grassmann's father Justus (1829, 1830) published two works on the geometrical description of crystals, influenced by the earlier works of C.S. Weiss (1780-1856) on three main crystal forces governing crystal formation. In…
We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple…
We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra $\mathbb{C}_{s}$. We exploit the observation that…
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean…
This paper develops a geometrical model of dislocations and disclinations in single crystals at the mesoscopic scale. In the continuation of previous work the distribution theory is used to represent concentrated effects in the defect lines…
The intimate link between complex geometry and the problem of the pre-metric formulation of electromagnetism is explored. In particular, the relationship between 3+1 decompositions of R4 and the decompositions of the vector space of…
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.
Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph…
In this paper we present some new, practical, geometric optimization techniques for computing polygon partitions, 1D and 2D point, interval, square and rectangle covers, as well as 1D and 2D interval and rectangle K-centers. All the…
We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. This way symmetry…
The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections…
The goal of this work is to study some aspects of the geometry of the first cover $\Sigma^1$ in the Drinfeld tower over $\mathbb{H}^d_K$ the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. It is a cyclic \'etale…
We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of…
For a Kac-Moody group $G$, double Bruhat cells $G^{u,e}$ ($u$ is a Weyl group element) have positive geometric crystal structures. In arXiv:1210.2533, it is shown that there exist birational maps between `cluster tori'…