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Axioms
Special Issues
Trends in Differential Geometry and Algebraic Topology
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Trends in Differential Geometry and Algebraic Topology

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: 30 November 2025 | Viewed by 3158

Special Issue Editor

Dr. Byungdo Park

E-Mail Website
Guest Editor
Department of Mathematics Education, Chungbuk National University, Cheongju 28644, Republic of Korea
Interests: differential geometry and algebraic topology; geometric structures on higher bundles; generalized cohomology theories and their equivariant, differential, and twisted refinements

Special Issue Information

Dear Colleague,

We are pleased to invite you to contribute to a Special Issue of Axioms titled "Trends in Differential Geometry and Algebraic Topology." This issue aims to showcase cutting-edge research and novel perspectives at the intersection of these two fundamental areas of mathematics.

As a recognized expert in the field, your contribution would be invaluable in shaping the discourse on recent developments and future directions. We welcome original research articles, comprehensive reviews, and insightful perspectives that explore the following:

  • Advances in differential geometric structures and their applications, including Riemannian and symplectic geometry;
  • Higher category theory, homotopy theory, homology theory, algebraic K-theory;
  • Interplay between differential geometry and algebraic topology;
  • Index theory, noncommutative geometry;
  • Emerging computational methods in geometric and topological analysis;
  • Applications of differential geometry and algebraic topology in physics, data science, or other disciplines.

We are excited about the possibility of featuring your work in this Special Issue and contributing to the advancement of these vital mathematical fields.

We look forward to the prospect of your valuable contribution.

Dr. Byungdo Park
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. Axioms 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 2400 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

  • differential geometry
  • algebraic topology
  • Riemannian geometry
  • symplectic geometry
  • symplectic topology
  • homotopy algebra
  • homotopy theory
  • homology theory
  • category theory
  • algebraic K-theory
  • topological K-theory
  • index theory
  • noncommutative geometry
  • string theory
  • quantum field theory

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  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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Research

15 pages, 276 KiB  
Article
Irresolute Homotopy and Covering Theory in Irresolute Topological Groups
by Kadriye Başar and Hürmet Fulya Akız
Axioms 2025, 14(4), 308; https://doi.org/10.3390/axioms14040308 - 17 Apr 2025
Viewed by 140
Abstract
In this paper, we explore certain properties related to connectedness and introduce the definition of irresolute paths. Subsequently, we define the concepts of semi-path connectedness, locally semi-path connectedness, and semi-locally s-simply connected spaces. Additionally, we introduce the concept of irresolute homotopy and reconstruct [...] Read more.
In this paper, we explore certain properties related to connectedness and introduce the definition of irresolute paths. Subsequently, we define the concepts of semi-path connectedness, locally semi-path connectedness, and semi-locally s-simply connected spaces. Additionally, we introduce the concept of irresolute homotopy and reconstruct the fundamental group based on this framework. Furthermore, we prove that the structure of an irresolute topological group with a universal irresolute covering can be lifted to its irresolute covering space. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
13 pages, 273 KiB  
Article
Filtered Products of Copies of Injective Modules
by Driss Bennis, Juan Ramón García Rozas, Maroua Mbarki and Luis Oyonarte
Axioms 2025, 14(4), 303; https://doi.org/10.3390/axioms14040303 - 16 Apr 2025
Viewed by 122
Abstract
We study the transfer of injectivity to filtered products of copies of an injective module. This leads to the introduction of a generalized Noetherian condition, the so-called (ℵ,M)-Noetherian rings. We prove that M is F-injective for [...] Read more.
We study the transfer of injectivity to filtered products of copies of an injective module. This leads to the introduction of a generalized Noetherian condition, the so-called (ℵ,M)-Noetherian rings. We prove that M is F-injective for every filter F with cpl(F)≥ℵ if and only if R is (ℵ,M)-Noetherian. We also examine the behavior of filtered products of τ-injective torsion-free modules, establishing preservation results under suitable conditions. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
49 pages, 471 KiB  
Article
Quasi-Elliptic Cohomology of 4-Spheres
by Zhen Huan
Axioms 2025, 14(4), 267; https://doi.org/10.3390/axioms14040267 - 1 Apr 2025
Viewed by 172
Abstract
It is a famous hypothesis that orbifold D-brane charges in string theory can be classified in twisted equivariant K-theory. Recently, it is believed that the hypothesis has a non-trivial lift to M-branes classified in twisted real equivariant 4-Cohomotopy. Quasi-elliptic cohomology, which is defined [...] Read more.
It is a famous hypothesis that orbifold D-brane charges in string theory can be classified in twisted equivariant K-theory. Recently, it is believed that the hypothesis has a non-trivial lift to M-branes classified in twisted real equivariant 4-Cohomotopy. Quasi-elliptic cohomology, which is defined as an equivariant cohomology of a cyclification of orbifolds, potentially interpolates the two statements, by approximating equivariant 4-Cohomotopy classified by 4-sphere orbifolds. In this paper we compute Real and complex quasi-elliptic cohomology theories of 4-spheres under the action by some finite subgroups that are the most interesting isotropy groups where the M5-branes may sit. The computation connects the M-brane charges in the presence of discrete symmetries to Real quasi-elliptic cohomology theories, and those with the symmetry omitted to complex quasi-elliptic cohomology theories. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
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26 pages, 495 KiB  
Article
Beyond Algebraic Superstring Compactification
by Tristan Hübsch
Axioms 2025, 14(4), 236; https://doi.org/10.3390/axioms14040236 - 21 Mar 2025
Viewed by 843
Abstract
Superstring compactifications have been vigorously studied for over four decades, and have flourished, involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are “purely” algebraic. Recent developments however indicate [...] Read more.
Superstring compactifications have been vigorously studied for over four decades, and have flourished, involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are “purely” algebraic. Recent developments however indicate many more possibilities to be afforded by including certain generalizations that, at first glance at least, are not algebraic—yet fit remarkably well within an overall mirror-symmetric framework and are surprisingly amenable to standard computational analysis upon certain mild but systematic modifications. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
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10 pages, 248 KiB  
Article
Net-Compact Hausdorff Topologies and Continuous Multi-Utility Representations for Closed Preorders
by Gianni Bosi, Gabriele Sbaiz and Magalì Zuanon
Axioms 2025, 14(3), 188; https://doi.org/10.3390/axioms14030188 - 3 Mar 2025
Viewed by 293
Abstract
In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we [...] Read more.
In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we show that if every closed preorder on a net-compact Hausdorff topological space has a continuous multi-utility representation, then the topology is normal. Second, we prove that every closed preorder on a normal and net-compact Hausdorff topological space admits a continuous multi-utility representation. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
14 pages, 292 KiB  
Article
Duality and Some Links Between Riemannian Submersion, F-Harmonicity, and Cohomology
by Bang-Yen Chen and Shihshu (Walter) Wei
Axioms 2025, 14(3), 162; https://doi.org/10.3390/axioms14030162 - 23 Feb 2025
Viewed by 501
Abstract
Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields [...] Read more.
Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields in physics are exactly the same. In n-dimensional geometry, a fundamental notion is the “duality” between chains and cochains, or domains of integration and the integrands. In this paper, we extend ideas given in our earlier articles and connect seemingly unrelated areas of F-harmonic maps, f-harmonic maps, and cohomology classes via duality. By studying cohomology classes that are related with p-harmonic morphisms, F-harmonic maps, and f-harmonic maps, we extend several of our previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic maps and f-harmonic maps, which are Riemannian submersions. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
12 pages, 267 KiB  
Article
An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points
by Javier Rodrigo, Mariló López, Danilo Magistrali and Estrella Alonso
Axioms 2025, 14(1), 62; https://doi.org/10.3390/axioms14010062 - 16 Jan 2025
Viewed by 512
Abstract
In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the [...] Read more.
In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 points, provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples, and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all n multiples of 3, there are 3-symmetric sets of n points for which the rectilinear crossing number is attained. Full article
(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)
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