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Mathematics
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Noncommutative Geometry and Number Theory
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Noncommutative Geometry and Number Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 April 2020) | Viewed by 8147

Special Issue Editor

Prof. Dr. Igor V. Nikolaev

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, New York, NY 11439, USA
Interests: noncommutative geometry; number theory

Special Issue Information

Dear Colleagues,

There is a growing evidence that noncommutative geometry may have a lasting impact on the unsolved classical problems of number theory; see the work of Bost and Connes on the Riemann Hypothesis, Cuntz's generalization of the Bost–Connes systems, and Manin's real multiplication program. The goal of the Special Issue is to advance in this direction by collecting articles related to the following concrete problems: (i) the Manin's approach to Hilbert's twelfth problem (“Kronecker's Jugendtraum") about the explicit construction of generators of the abelian extensions of the real quadratic fields; (ii) a revision of the Weil's conjectures using the trace cohomology coming from the K -theory of operator algebras; (iii) and to recast and understand the Langlands conjectures in terms of the operator algebras. The methods are an interplay between the operator algebras (Serre C*-algebras and non-commutative tori), algebraic geometry (abelian varieties and complex multiplication), and number theory (rational elliptic curves).

Prof. Igor V. Nikolaev
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 submissions that pass pre-check are 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.

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 semimonthly 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 2600 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

  • Noncommutative torus
  • Real multiplication
  • K-theory
  • Elliptic curve
  • Complex multiplication
  • Hilbert’s twelfth problem
  • Weil conjectures
  • Langlands program

Published Papers (4 papers)

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Research

14 pages, 294 KiB  
Article
A Note on Symmetry of Birkhoff-James Orthogonality in Positive Cones of Locally C*-algebras
by Alexander A. Katz
Mathematics 2020, 8(6), 1027; https://doi.org/10.3390/math8061027 - 23 Jun 2020
Cited by 1 | Viewed by 1688
Abstract
In the present note some results of Kimuro, Saito, and Tanaka on symmetry of Birkhoff-James orthogonality in positive cones of C*-algebras are extended to locally C*-algebras. Full article
(This article belongs to the Special Issue Noncommutative Geometry and Number Theory)
11 pages, 239 KiB  
Article
Ideals on the Quantum Plane’s Jet Space
by Andrey Glubokov
Mathematics 2020, 8(3), 352; https://doi.org/10.3390/math8030352 - 06 Mar 2020
Viewed by 1329
Abstract
The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (Theorems 1–4) about the prime [...] Read more.
The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (Theorems 1–4) about the prime spectrum of these rings. Full article
(This article belongs to the Special Issue Noncommutative Geometry and Number Theory)
10 pages, 256 KiB  
Article
On Expansive Mappings
by Marat V. Markin and Edward S. Sichel
Mathematics 2019, 7(11), 1004; https://doi.org/10.3390/math7111004 - 23 Oct 2019
Cited by 1 | Viewed by 3044
Abstract
When finding an original proof to a known result describing expansive mappings on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is [...] Read more.
When finding an original proof to a known result describing expansive mappings on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is found showing that the converse to the above descriptions do not hold, we are able to characterize boundedness in terms of specific expansions we call anticontractions. Full article
(This article belongs to the Special Issue Noncommutative Geometry and Number Theory)
8 pages, 219 KiB  
Article
On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators
by Marat V. Markin and Edward S. Sichel
Mathematics 2019, 7(10), 903; https://doi.org/10.3390/math7100903 - 27 Sep 2019
Cited by 3 | Viewed by 1581
Abstract
We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition [...] Read more.
We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators. Full article
(This article belongs to the Special Issue Noncommutative Geometry and Number Theory)
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