Differential Geometry and Its Applications in Theoretical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 20 October 2026 | Viewed by 984

Editor


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Guest Editor
Institute of Mathematics and Statistics, University of Tartu, 51009 Tartu, Estonia
Interests: differential geometry; gauge theories; supermanifolds; n-Lie algebras; superalgebras

Special Issue Information

Dear Colleagues,

The aim of this Special Issue of Mathematics is to highlight current trends in the development of differential geometry, to present newly developed structures, and to discuss the methods used in their study. An important part of the project is the broad area of applications of differential geometry in theoretical physics. The mutual influence of modern algebra and differential geometry is also an essential aspect of this project. Research in the following thematic areas falls within the scope of the Special Issue, although it is not limited to them:

  1. Connections in fiber bundles, gauge theories, and their generalizations;
  2. Lie groups, Lie algebras, Poisson algebras, and their generalizations;
  3. Riemannian geometry and general relativity;
  4. Noncommutative geometry;
  5. Geometric structures in string theory and M-theory;
  6. Ternary algebras and quantum physics.

Prof. Dr. Viktor Abramov
Guest Editor

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Keywords

  • differential geometry
  • algebras
  • general relativity
  • theoretical physics

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Published Papers (1 paper)

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Research

28 pages, 518 KB  
Article
Geometry of Deformed Cellular Spaces
by Shlomo Barak and George Salman
Mathematics 2026, 14(11), 1824; https://doi.org/10.3390/math14111824 - 24 May 2026
Viewed by 478
Abstract
We develop an operational, measurement-first framework for the geometry of locally finite cell complexes, in which length is defined as a count of face crossings, and curvature is read off from the discrepancy between a measured radius and a radius reconstructed from boundary, [...] Read more.
We develop an operational, measurement-first framework for the geometry of locally finite cell complexes, in which length is defined as a count of face crossings, and curvature is read off from the discrepancy between a measured radius and a radius reconstructed from boundary, area, or volume counts using the same yardstick. We prove that the count metric is geodesic on every locally finite complex, and we introduce a unified small-ball/small-sphere curvature estimator that is valid in dimensions two through four with a single closed-form expression. By comparison with the standard small-ball volume expansion of a smooth conformal metric g=e2ug0, we establish a quantitative identification theorem with explicit rate O(a/r+r), which optimizes to O(a) at ra. We extend the construction to directional (sectional) estimators via Fermi tubes around geodesic two-slices, assemble the curvature operator, Ricci tensor, and scalar curvature in three dimensions, and prove a measured Gromov–Hausdorff convergence theorem for the rescaled count metric. All hypotheses are verified explicitly on Voronoi complexes of conformal metrics. Throughout, we are explicit that the discrete construction is interpreted via, and its asymptotic validity is established by comparison with, the smooth Riemannian theory; the contribution is the unified counts-only protocol with rigorous convergence rates, not a reformulation of curvature itself. Full article
(This article belongs to the Special Issue Differential Geometry and Its Applications in Theoretical Physics)
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