Special Issue "Mirror Symmetry and Algebraic Geometry"

A special issue of Symmetry (ISSN 2073-8994).

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

Special Issue Editor

Prof. Dorin Popescu
Website
Guest Editor
Senior Researcher I in Simion Stoilow Institute of Mathematics of Romanian Academy and Professor in the Doctoral School of Department of Mathematics of University of Bucharest
Interests: commutative algebra and algebraic geometry

Special Issue Information

Dear Colleagues,

The Galois group of a polynomial f with integral coefficients is a measure of the symmetry of the complex roots of f and the solvability of the equation f = 0 by radicals, it is just a question of the symmetry of the roots of f.

The mirror symmetry leads the physicists to do important predictions about the rational curves on the quintic threefold, which were partially proved very late by people from Algebraic Geometry. The prediction about Gromov–Witten invariants given by the mirror symmetry is now proved mathematically in several cases. Many new fields and concepts in Algebraic Geometry appeared when people tried to give a mathematical foundation for aspects of the mirror symmetry, for example, quantum cohomology, the complexified Kahler moduli space of a Calabi–Yau threefold, Kontsevich's definition of a stable map, and Batyrev's duality between certain toric varieties and Givental's notion of Quantum Differential Equations.

Prof. Dorin Popescu
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Keywords

  • Mirror symmetry
  • Rational curve
  • Threefold
  • Moduli space
  • Toric variety
  • Gromov–Witten invariants
  • Quantum cohomology

Published Papers (4 papers)

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Research

Open AccessArticle
Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure
Symmetry 2019, 11(7), 934; https://doi.org/10.3390/sym11070934 - 17 Jul 2019
Cited by 1
Abstract
We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in [...] Read more.
We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions. Full article
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
Open AccessArticle
Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups
Symmetry 2019, 11(7), 902; https://doi.org/10.3390/sym11070902 - 11 Jul 2019
Abstract
The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order [...] Read more.
The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space. Full article
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
Open AccessArticle
Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras
Symmetry 2019, 11(7), 881; https://doi.org/10.3390/sym11070881 - 04 Jul 2019
Cited by 1
Abstract
In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and [...] Read more.
In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series. Full article
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
Open AccessArticle
SAGBI Bases in G-Algebras
Symmetry 2019, 11(2), 221; https://doi.org/10.3390/sym11020221 - 13 Feb 2019
Abstract
In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm [...] Read more.
In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm to compute SAGBI bases from a subset of polynomials contained in a subalgebra of a G-algebra. Full article
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
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