Abstract: Unlike geometry, spheres in topology have been seen as topological invariants, where their structures are defined as topological spaces. Forgetting the exact notion of geometry, and the impossibility of embedding one into another, homotopy theory relates how a sphere of one dimension can wrap around, or map into, a sphere of another dimension. This paper revisits the classical theory of homotopy groups of spheres, providing a detailed exploration of their computation and structure. We place special emphasis on the pivotal role of Hopf fibrations in revealing the higher homotopy groups of spheres, particularly the exotic and fascinating case of π₃(S²). Furthermore, we explore the elegant geometric connection to Villarceau circles, demonstrating how these circles on a torus are intimately linked to the Hopf fibration of S³. This work serves as a comprehensive guide, bridging abstract algebraic topology with tangible geometric phenomena. This version expands significantly on the foundational ideas, providing deeper insights and connections to contemporary research, including stable homotopy theory, the Adams conjecture, and generalizations to Calabi-Yau manifolds. (An earlier version of this work was published as: D. Bhattacharjee, S. Singha Roy, R. Sadhu, “Homotopy group of spheres, Hopf fibrations and Villarceau c ircles”, EPRA International Journal of Research & Development (IJRD), Vol. 7, Issue 9, September 2022. DOI: https://doi.org/10.36713/epra11212)

Abstract: This study investigates the dripping and leakage problem in kitchenware known as the "teapot effect" through a multidisciplinary experimental approach encompassing fluid mechanics, material science, and ergonomic design. Unlike previous studies confined to idealized geometries and single-fluid analyses, this work systematically examines 32 distinct spout geometries from commercially available teapots, coffee pots, and milk jugs under realistic operating conditions. Experiments were performed using four fluids with contrasting rheological properties water, boiling black tea, cow's milk, and Turkish coffee on a precision rotating platform operating at 1°/s to isolate surface tension, gravitational, and geometric effects from inertial forces. Three quantitative parameters were measured for each specimen: capillary dome angle, teapot effect angle range, and optimum pouring angle. Results demonstrate that spout tip geometry is the dominant controlling parameter. Thin-lipped elliptical cross-sections effectively suppressed dripping, whereas triangular and wide curved geometries produced the teapot effect across broad pouring angle ranges reaching up to 70°. A spout outlet extension length of 4–5 mm combined with a spout tip radius below 4 mm was found necessary and sufficient for clean flow separation. Furthermore, suspended particles and proteins in milk and Turkish coffee were shown to intensify the teapot effect by disrupting contact line dynamics at the spout tip. These findings provide quantitative design thresholds directly applicable to industrial kitchenware development.

Abstract: The aim of the article is to introduce a few variants of generalized quasi-continuity of multifunctions defined on a bitopological space and to study their mutual relationship. The results known for functions are extended to multifunctions which provide a wider range of relationships, mainly in terms of upper and lower semi continuities and corresponding continuities with respect to a dual bitopology. The proof procedures are based on a notion of pseudo refinement of two topologies and the Baire property in a bitopological space. A characterization of some continuities depending on two topologies by continuities depending only on one topology and the structure of the sets of semi discontinuity points are given. The end of the article is dedicated to several interpretations that facilitate and clarify orientation in the achieved results.

Abstract: This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are existence and uniqueness of hexagonal packings, in which each sphere is tangent to six others. Constructions are both intuitive and subtle, but result in the complete characterization in term of integer parameter pairs $(m,n)$. Interesting questions in rigidity and density are encountered. Density questions arise because the packings, being of equal diameter, lie within the space between inner and outer cylinders. This density problem hoovers between the 2D and 3D sphere packing cases, and though it is not solved here, it is conjectured that the hexagonal packings are densest for the countable number of cylinders which support them. Other geometric objects are along for the ride, including equilateral triangles and the packings' dual graphs, which are associated with patterns of carbon atoms forming buckytubes. Interesting structural rigidity questions also arise.

Abstract: We study involutions on complex K3 surfaces and their quotients, focusing on the emergence of Enriques and Kummer surfaces. Emphasis is placed on lattice-theoretic structures, geometric invariants, and projective realizations via the Kodaira embedding theorem. Examples are provided to illustrate the relationships among these surface classes and their induced geometric properties.

Abstract: This work investigates a mechanism for topology-altering geometric behavior induced by angular group actions within the framework of the Alpha Group, demonstrating that a systematic angular sweep can drive spontaneous transitions between Riemannian-like and sub-Riemannian regimes. Motivated by the group-theoretic foundations of geometry established by the Göttingen school and by the central role of invariants in topology, we introduce a setting in which geometric organization emerges from the interplay between an invariant, idempotent operator µ and an angular matrix M(θ ). While µ provides a stable structural background, the angular action associated with M(θ ) induces anisotropy and directional dependence. A systematic angular sweep is performed to probe the geometric response of the in- duced structure, revealing the spontaneous emergence of distinct isotropic, moderate, and strong regimes. These regimes are not imposed a priori, but arise dynamically from the interaction between the invariant background and the angular action. As the angular pa- rameter departs from stable configurations, anisotropic effects activate a sub-Riemannian Carnot–Carathéodory geometric framework, leading to a qualitative reorganization of the underlying topology. The strong regime is characterized by localized and persistent geometric features, ac- tivated only within bounded angular intervals and producing symmetric bifurcation pat- terns around θ = 90◦ . Despite these anisotropic reorganizations, global coherence and connectivity are preserved by the invariant action of µ. These results demonstrate that sub-Riemannian geometry can act as a natural driver of dynamic topological restructur- ing within group-based geometric frameworks, providing a coherent alternative to classical Riemannian descriptions.

Abstract: We investigate a potential route to the Riemann Hypothesis based on de Branges positivity and wild isomonodromic geometry, focusing on Painlevé III of type D6. Rather than proposing a proof, we reduce any such route to four explicit conditions (C1)–(C4), isolating a single analytic bottleneck: the existence of a global positivity normalization for the associated wild Riemann–Hilbert problem. Using the decorated character variety framework of Chekhov–Mazzocco–Rubtsov and the embedding t = s(1 − s), we show that symmetry, gauge freedom, and growth constraints of the completed zeta function are all compatible with this setting. We further perform a quantitative density test based on the Weyl–Levinson law for canonical systems, showing that the zeta-induced spectral growth is highly selective yet not excluded by the Painlevé III_D6 Hamiltonian. The result is a falsifiable and discriminating framework that identifies where a de Branges-based realization of the Riemann Hypothesis must succeed or fail. We further analyze the analytic regularity condition (C4), show that the symmetry-compatibility condition (C3) is automatically satisfied for the natural embedding t = s(1− s), and isolate the global positivity condition (C2) as the decisive remaining analytic obstacle. In particular, we reduce (C2) to the absence of a single explicit Weyl–Herglotz obstruction for the associated canonical system, and develop falsifiable diagnostics, including a quantitative density test based on Weyl–Levinson asymptotics.

Abstract:

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are \( \textit{gauge structures on these vector bundles} \). We are interested in dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e. properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection affine connection? (P2-Riemannian Geometry): When is a Koszul connection metric connection? (P3-Fedosov Geometry): When is a Koszul connection symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of \( \textit{how to produce labeled foliations the most studied of which are Riemannian foliations} \). On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemeted to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce \( \textit{Koszul Homological Series}\textit{Koszul Homological Series} \). This notion is a machine for converting Obstructions whose nature is vector space into Obstructions whose nature is Homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2) and (P3). \( \textit{In the abundant literature on Riemannian foliatins we have only cited references directely related to the open problems which are studied using the tools which are introduced in this work. Thus the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How to produce Riemanian foliations?} \). See our Theorem 7.4 and Theorem 7.5 which are fruits of a happy conjunction between the gauge geometry and the differential topology.

Abstract:

We study a purely local inverse problem for non-reversible Randers metrics \( F = \|\cdot\|_g + \beta \) defined on smooth oriented surfaces. Using only the lengths of sufficiently small closed curves around a point \( p \), we prove that the exterior derivative \( d\beta(p) \) can be uniquely and stably recovered. Moreover, we establish that \( d\beta(p) \) is the only second-order local invariant retrievable from such local length measurements. Our approach is entirely metric-based, independent of geodesic flows or boundary data, and naturally extends to general curved surfaces.

Abstract: We present the Atemporal Tablet Framework (ATF), a complete geometric ontology that derives spacetime, quantum mechanics, and gravity from a single mathematical structure. The universe is modeled as a fiber bundle T ->(π) M where T is a static higher-dimensional manifold and M is emergent 3+1D spacetime. Temporal dynamics arise from projection operators Πt : T -> M extremizing a projective action SΠ. Quantum states are epistemic distributions over fibers, with the Born rule emerging naturally via measure disintegration. Measurement corresponds to topological phase-locking without wavefunction collapse. Einstein’s equations arise as equations of motion for Πt, while quantum fields emerge as fiber vibrations. The framework makes specific testable predictions: sidereal anisotropy in qubit decoherence ε = 1.23 × 10^-8 ± 3 × 10^-9 (derived from holographic scaling) and modified dispersion relations at scale EP / sqrt(ε). We prove a reconstruction theorem establishing that spacetime observations can determine the underlying geometry, and demonstrate that Standard Model particle content emerges naturally from Fx ≅ CP3 × S5 / Γ fiber geometry. ATF provides a mathematically rigorous, experimentally falsifiable foundation for quantum gravity that resolves long-standing interpretational issues while making concrete predictions testable with current technology.

Abstract: Each of the studied manifolds has a pair of B-metrics, interrelated by an almost contact structure. The case where each of these metrics gives rise to an η-Ricci–Bourguignon almost soliton, where η is the contact form, is studied. In addition, the geometry-rich case where the soliton potential is torse-forming and is pointwise collinear on the Reeb vector field with respect to each of the two metrics is considered. Ricci tensors and scalar curvatures are expressed as functions of the parameters of the pair of almost solitons. Particular attention is paid to the special case when the manifold belongs to the only possible basic class of the corresponding classification. A necessary and sufficient condition has been found for these almost solitons to be η-Einstein for both metrics.

Abstract: This paper presents a definitive synthetic proof of the impossibility of trisecting an arbitrary angle within Euclidean geometry. The proof centers not on algebraic abstractions, but on an intrinsic geometric inconsistency revealed through the lens of the canonical 90° angle. This angle serves not merely as a counterexample, but as a diagnostic lever that fractures the very concept of a universal trisection property. A new “Principle of Operational Dissonance” is formulated from an analysis of foundational operations, such as doubling and cubing a square’s diagonal. These operations, while producing congruent final magnitudes, violate the core Euclidean doctrine of proportional similarity, demonstrating that (a:b ≠ c:d) in a strict geometric sense. This dissonance mirrors the logical structure of the trisection problem. The proof demonstrates that assuming the existence of a universal trisection procedure forces a specific geometric condition-the equality of certain lengths-when applied to a 90° angle. This condition arises solely from the angle’s axiomatic status and the constraints of compass-and-straightedge constructions. However, this forced condition is not preserved under variation of the angle measure, rendering any purported universal procedure internally inconsistent. The resulting contradiction proves the impossibility of trisecting a 90° angle with a universal method. This failure, stemming from a fundamental incompatibility within the geometric system rather than the peculiarities of a single angle, extends to all angles, conclusively resolving the classical problem. The proof thus delineates the exact boundary of classical constructive geometry, indicating that any future universal solution must arise from the introduction of new geometric properties innately compatible with Euclidean theory. It reaffirms the self-contained sufficiency of Euclidean geometry for resolving its celebrated problems and challenges the methodological necessity of importing non-geometric techniques to establish geometric impossibilities. The presented framework offers a purely synthetic geometric perspective, one that aligns with the foundational spirit of Euclid’s Elements.

Abstract: We study the local recovery of magnetic invariants in smooth n-dimensional manifolds equipped with general non-reversible Finsler metrics. We prove that the exterior derivative dβ is the unique second-order antisymmetric local invariant of the length functional, independently of higher-order Finsler perturbations. This generalizes previous 2-dimensional results to higher dimensions and establishes a rigorous, practically stable procedure for isolating magnetic invariants locally.

Abstract:

We establish a perturbative stability result for the Yamabe problem under genuinely non-conformal tensorial deformations on closed Riemannian manifolds \( (M^n,g) \), \( n \ge 3 \), incorporating small symmetric perturbations \( T \in C^{2,\alpha}(S^2 T^* M) \) beyond classical conformal rescalings. By introducing a tensorial correction \( E[T,\nabla T] \) in the scalar curvature functional, we define a perturbed variational problem whose critical points satisfy the modified Euler--Lagrange equation \( -a \Delta_g \Omega + (R_g + E[T,\nabla T]) \Omega = \lambda \, \Omega^p, \quad p = \frac{n+2}{n-2} \). Using precise linearization of Ricci and scalar curvatures, we derive quantitative estimates for the trace-free Ricci contributions and prove a rigidity theorem on Einstein backgrounds: sufficiently small perturbations T cannot generate a nontrivial trace-free Ricci component. Moreover, we establish perturbative existence and uniqueness of solutions \( \Omega_T \) in the perturbed conformal class \( \mathcal{C}(g,T) \), with explicit control \( \|\Omega_T - \Omega_0\|_{C^{2,\alpha}} \le C \, \|T\|_{C^{2,\alpha}} \). Our analysis provides a rigorous framework connecting classical Yamabe theory to tensorial deformations, yielding sharp stability estimates, bounds on linear and higher-order contributions, and explicit conditions under which Einstein metrics remain locally rigid. These results form a foundation for future investigations on higher-order curvature operators, large perturbations, and bifurcation phenomena in generalized Yamabe-type problems.

Abstract: We extend the theory of interdimensional weak null-preserving maps by provid- ing a complete local classification of the deviation tensor T based on its rank and kernel. We define weak k-plane null-preserving maps, examine their canonical de- composition, and analyze the local stability of T under small perturbations. Explicit examples illustrate the new classes of local behaviors. These results offer a rigor- ous and original contribution to the study of null structures in pseudo-Riemannian geometry.

Abstract:

We construct a local framework for maps between pseudo-Riemannian manifolds of different dimensions that preserve null directions. Let \( F: M_i \to M_{i+1} \); F is null-preserving if \( F_* v \) is null for every null \( v \in T_p M_i \). Deviations from an exact metric pullback are measured via a correction tensor T. This setup extends Liouville-type uniqueness results to the interdimensional case, providing a precise tool for local analysis of geometric embeddings.

Abstract: In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces, which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of θs-completeness and, as an application of the main theorems, we derive some results in both quantale-valued partial metric spaces and probabilisic metric spaces.

Abstract: Complex networks model the structure and function of critical technological, biological, and communication systems. Network dismantling, the targeted removal of nodes to fragment a network, is essential for analyzing and improving system robustness. Existing dismantling methods suffer from key limitations: they depend on global structural knowledge, exhibit slow running times on large networks, and overlook the network’s latent geometry, a key feature known to govern the dynamics of complex systems. Motivated by these findings, we introduce Latent Geometry-Driven Network Automata (LGD-NA), a novel framework that leverages local network automata rules to approximate effective link distances between interacting nodes. LGD-NA is able to identify critical nodes and capture latent manifold information of a network for effective and efficient dismantling. We show that this latent geometry-driven approach outperforms all existing dismantling algorithms, including spectral Laplacian-based methods and machine learning ones such as graph neural networks and . We also find that a simple common-neighbor-based network automata rule achieves near state-of-the-art performance, highlighting the effectiveness of minimal local information for dismantling. LGD-NA is extensively validated on the largest and most diverse collection of real-world networks to date (1,475 real-world networks across 32 complex systems domains) and scales efficiently to large networks via GPU acceleration. Finally, we leverage the explainability of our common-neighbor approach to engineer network robustness, substantially increasing the resilience of real-world networks. We validate LGD-NA's practical utility on domain-specific functional metrics, spanning neuronal firing rates in the Drosophila Connectome, transport efficiency in flight maps, outbreak sizes in contact networks, and communication pathways in terrorist cells. Our results confirm latent geometry as a fundamental principle for understanding the robustness of real-world systems, adding dismantling to the growing set of processes that network geometry can explain.

Abstract: Classical loop-spaces capture cyclic behaviour in topology but are blind to the auxiliary data that often drives real-world quasi-periodic phenomena. In this paper we introduce decorated loop-spaces, organised into a category $\mathbf{DecLpSpc}$, whose objects are spaces equipped with “decorators” (labelling generators by auxiliary data) and whose morphisms are “connectors” acting on families of functions. We construct a decorated loop functor $$\widehat{\Omega} : \mathbf{DecLpSpc} \to \mathbf{DecLpSpc},$$ define a notion of decorated concatenation, and prove coherence and functoriality results in the spirit of Eckmann–Hilton duality. On the homotopical side, we extend classical Whitehead products and higher homotopy brackets to the decorated setting, obtaining decorated Whitehead products and Jacobiators that refine the quasi-Lie structure on homotopy groups by keeping track of decoration data. We show that $\mathbf{DecLpSpc}$ admits a natural symmetric monoidal structure and support operads acting on decorated loop-spaces, giving a recognition principle for iterated decorated loop functors $\widehat{\Omega}^n$. A worked example on a wedge of spheres illustrates how decorations enrich a nontrivial Whitehead product with additional algebraic labels. Finally, we outline several applications in which decorations encode physically or computationally meaningful structure: string dynamics and vacuum expectation values in background fluxes, evolutionary dynamics where decorations separate epigenetic from phenotypic data, and feedback and signal-processing architectures (including an OCR-inspired case study) where connectors transport function families between different feature spaces. We conclude with directions for an intrinsic homotopy theory of $\mathbf{DecLpSpc}$, computable invariants, and data-driven variants of the framework.

Abstract: Graph-structured data underpin a wide range of real-world systems, from molecular chemistry and biological interactions to social and information networks. Recent advancements in deep generative models, along with the emergence of large language models (LLMs), have spurred significant progress in graph generative models (GGMs), enabling the synthesis of complex and realistic graph structures. This article provides a comprehensive overview of the literature in this field. We begin by distinguishing between two primary categories of graph datasets based on their structural formation mechanisms: geometric and scale-free. Building on this foundation, we propose a unified taxonomy that systematically organizes this rapidly evolving landscape by jointly considering (i) the category of generated graphs, (ii) the graph attribute modality, and (iii) the underlying probabilistic modeling paradigm. We then analyze representative neural network architectures and modeling strategies, followed by an overview of evaluation metrics. Finally, we highlight key applications in molecular design, protein optimization, social network analysis, and recommendation systems and outline four promising directions for future research.