Special Issue "Advances in singularities throughout Mathematics: Algebra, Topology, Geometry, Analysis, Number Theory, Combinatorics and Cryptography"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2019).

Special Issue Editor

Prof. José Ignacio Cogolludo-Agustín
Website
Guest Editor
Departamento de Matemáticas, IUMA, Universidad de Zaragoza, Zaragoza, Spain
Interests: singularities; algebraic geometry; algebraic combinatorics; algebraic curves; topology of algebraic varieties; polytopes and lattice point counting

Special Issue Information

Dear Colleagues,

The increasing amount of new material related to singularity theory from a broad perspective is an opportunity to approach mathematics from this transversal viewpoint. From group theory to differential geometry, from topology to cryptography, from algebraic geometry to number theory or analysis, elements and topics in singularity theory establish connections that are both surprising and enriching.

The purpose of this Special Issue is to give researchers an opportunity to present their achievements in a monograph that offers an overview of the different perspectives, techniques, and applications of the modern and classical theory of singularities.

Possible topics of this issue are related, but not restricted, to the following:

  • Topology of singular varieties;
  • Local vs. global invariants of singularities: Normal singularities, zeta functions, Poincaré and Hilbert-type series, multiplier ideals, D-modules and B-polynomials, etc.
  • Theory of hyperplanes: Hyperplane arrangements, toric hyperplanes, modules of logarithmic derivations, Terao’s conjecture, etc.
  • Singular dynamical systems, orbits, stability, periodic solutions, etc.
  • Polynomial maps: Jacobian conjecture, Milnor fibrations, elliptic fibrations, etc.
  • Motivic invariants: Grothendieck ring, space of arcs, Nash problem, etc.
  • Singular maps in differential geometry: Classification, bifurcation theory, etc.
  • Deformation theory;
  • Real and complex structures: Hodge theory, plurigenera, formality, etc.
  • Groups in singularity theory: Quasiprojective or Kähler groups, braid monodromy, quotient singularities, invariant theory, etc.
  • Combinatorial aspects: Toric varieties, lattice point counting, etc.
  • Cryptography: Toric and polynomial maps, multivariate systems, etc.

Prof. José Ignacio Cogolludo-Agustín
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Singular varieties
  • Singular maps
  • Hodge structures
  • Singular dynamical systems
  • Theory of hyperplanes
  • Toric varieties
  • Real and complex singularities
  • MS cryptography

Published Papers (3 papers)

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Research

Open AccessFeature PaperArticle
Singularities of Non-Developable Surfaces in Three-Dimensional Euclidean Space
Mathematics 2019, 7(11), 1106; https://doi.org/10.3390/math7111106 - 14 Nov 2019
Abstract
We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By [...] Read more.
We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By using structure functions, we can characterize the ruled surface generated by special curves. Full article
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Open AccessArticle
Geometric Characterizations of Canal Surfaces in Minkowski 3-Space II
Mathematics 2019, 7(8), 703; https://doi.org/10.3390/math7080703 - 05 Aug 2019
Abstract
Canal surfaces are defined and divided into nine types in Minkowski 3-space E 1 3 , which are obtained as the envelope of a family of pseudospheres S 1 2 , pseudohyperbolic spheres H 0 2 , or lightlike cones Q 2 , [...] Read more.
Canal surfaces are defined and divided into nine types in Minkowski 3-space E 1 3 , which are obtained as the envelope of a family of pseudospheres S 1 2 , pseudohyperbolic spheres H 0 2 , or lightlike cones Q 2 , whose centers lie on a space curve (resp. spacelike curve, timelike curve, or null curve). This paper focuses on canal surfaces foliated by pseudohyperbolic spheres H 0 2 along three kinds of space curves in E 1 3 . The geometric properties of such surfaces are presented by classifying the linear Weingarten canal surfaces, especially the relationship between the Gaussian curvature and the mean curvature of canal surfaces. Last but not least, two examples are shown to illustrate the construction of such surfaces. Full article
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Open AccessArticle
Quasi Semi-Border Singularities
Mathematics 2019, 7(6), 495; https://doi.org/10.3390/math7060495 - 01 Jun 2019
Abstract
We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard [...] Read more.
We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser’s homotopy technique. In addition, we adopt the version of Arnold’s spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section. Full article
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