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We develop a new method to construct explicit, regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely we show that for a large class of planar curves $(x(t), y(t))$…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By…
We study the bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors.…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane…
Toric B\'ezier patches generalize the classical tensor-product triangular and rectangular B\'ezier surfaces, extensively used in $CAGD$. The construction of toric B\'ezier surfaces corresponding to multi-sided convex hulls for known…
In this paper, we consider the higher Br\'ezin--Gross--Witten tau-functions, given by the matrix integrals. For these tau-functions we construct the canonical Kac--Schwarz operators, quantum spectral curves, and $W^{(3)}$-constraints. For…
We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks…
These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe…
Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In \cite{Mesnager9}, Mesnager \emph{et al.} introduced generalized plateaued functions in order to…
All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal…
We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to…
The notion of geometric k-normality for curves is introduced in complete generality and is investigated in the case of nodal and cuspidal curves living on several types of surfaces. We discuss and suggest some applications of this notion to…
We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is…
This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of…
We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…
The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for…