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: 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.

Abstract: We develop a topological-combinatorial framework applying classical Ramsey theory to systems of arcs connecting points on Jordan curves and their higher-dimensional analogues. A Jordan curve Λ partitions the plane into interior and exterior regions, enabling a canonical two-coloring of every arc connecting points on Λ according to whether its interior lies in Int(Λ) or Ext(Λ). Using this intrinsic coloring, we prove that any configuration of six points on Λ necessarily contains a monochromatic triangle, and that this property is invariant under all homeomorphisms of the plane. Extending the construction by including arcs lying on Λ itself yields a natural three-coloring, from which the classical value R(3,3.3)=17 guarantees the appearance of monochromatic triangles for sufficiently large point sets. For infinite point sets on Λ, the infinite Ramsey theorem ensures the existence of infinite monochromatic cliques, which we likewise show to be preserved under arbitrary topological deformations. The framework extends to Jordan surfaces and Jordan–Brouwer hypersurfaces in higher dimensions, where interior, exterior, and boundary regions again generate canonical colorings and Ramsey-type constraints. These results reveal a general principle: the separation properties of codimension-one topological boundaries induce universal combinatorial structures - such as monochromatic triangles and infinite monochromatic subsets - that are stable under continuous deformations. The approach offers new links between geometric topology, extremal combinatorics, and the analysis of constrained networks and interfaces.

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 machine learning methods such as graph neural networks. 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. 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: In this paper we suggest that the law of excluded middle may not be valid for the brain's logical system. These rules are hidden in the rules of combining the brain's information carrying signals. But, as we show,these rules can be studied for signals generated in a topological surface network model of the brain with surface spin-half particles. This network can generate all observed brain-like excitations and has two topological features. One comes from the fact that it exactly captures the topological connectivity of the brain and thus has complex connectivity, and the other from the presence of surface spin-half particles. These two topological features can be represented by two topological numbers. Signals are produced in the network by subunits, in response to local topology changing surface deformations, that are input signals. The deformations reduce the topology of the subunit to that of a sphere. The signals thus produced carry the topology numbers of the subunit as well as the deformation parameter values that create them. They carry information. The network is represented by a Riemann surface of large unknown genus with spin structure and the signals produced are described by Riemann theta functions. The spin topology number carried by a signal is expressed in terms of the theta characteristics of the Riemann theta function. The key idea is that the rules for combining signals is related to the rules of combining theta characteristics of the Riemann theta functions discovered by Frobenius. A theoretical argument, using Frobenius's results, shows that whether the law of excluded middle holds or not can be decided by a predicted observational differences between signals that represent a proposition and its converse. The observational result shows that the law of excluded middle is not valid for the brain.

Abstract: A polarity of an exceptional geometry of type E₆ is called regular if its fix structure, viewed as simplicial complex, is a building. Polarities which do not act trivial on the underlying field were classified a long time ago by Jacques Tits. In the present paper, we classify the regular polarities of exceptional geometries of type E₆ that act trivial on the underlying (arbitrary) field. As a result, we discover new subgeometries of the exceptional geometry of type E₆.

Abstract: The purpose of this paper is to determine the local dimension of the critical locus of a generic singularity. We use combinatorial methods to calculate this dimension in terms of a convex object associated with the singularity, called the Newton polyhedron. In the article we prove that the local dimension of the critical locus of a generic singularity \( f:(\mathbb{C}^n,0)\longrightarrow (\mathbb{C},0), n\leq 4, \) is equal to the combinatorial dimension of the Newton polyhedron of the gradient mapping \( \nabla f. \) Therefore there is some symmetry between combinatorial properties of the Newton polyhedron of a generic singularity and geometric properties of its critical locus.

Abstract: We present two propositions, each with a proof, and a theorem to establish a foundational framework for a novel perspective on quantum information framed in terms of differential geometry and topology. In particular, we show that the mapping to $\mathbb{S}^0$ naturally encodes the binary outcomes of entangled quantum states, providing a minimal yet powerful abstraction of quantum duality. Building on this, we introduce the concept of a \emph{discrete fiber bundle} to represent quantum steering and correlations, where each fiber corresponds to the two possible measurement outcomes of entangled qubits. This construction offers a new topological viewpoint on quantum information, distinct from traditional Hilbert-space or metric-based approaches. The present work serves as a preliminary formulation of this framework, with further developments to follow.

Abstract: Humans typically describe spatial location using names, hierarchies, and relative positions (e.g., east of, inside), yet mainstream GIS represents places primarily through geometric coordinates, rendering qualitative spatial queries computationally challenging. We introduce the Qualitative Place Model (QPM), a GIS-native framework that transforms standard boundary datasets and place layers into structured knowledge bases of Qualitative Place Description Qualitative Place Description (QPDs). QPM provides a homogeneous representation whereby administrative units and physical places are treated uniformly as Place entities. The model materializes a compact set of local relations—hierarchical containment, directional neighbourhood, and optional proximity—that support rich inferences through sound logical operations (inverse relationships and per-predicate transitive closure). We implement QPM as an ArcGIS Pro toolbox that computes and persists QPDs within a geodatabase, with optional RDF/GeoSPARQL export for SPARQL querying. This implementation enables natural-language-style spatial queries such as "Where is \( x \)?" or "Which places are north of \( x \)?" within standard GIS workflows. Evaluation on Wales (UK) administrative, postal, and electoral hierarchies plus a comprehensive place layer demonstrates robust performance: QPM generated 95.8% of expected basic-place statements (52,821 places) and achieved 89.7–96.5% coverage across administrative hierarchies. All QPDs proved unique under our deterministic signature. Despite compact storage requirements, directional relations expand by more than an order of magnitude (10.6× overall expansion) under logical closure, demonstrating substantial inferential power from a minimal stored representation. QPM complements geometric GIS with an explainable qualitative layer that aligns with human spatial cognition while remaining fully operational within standard GIS environments.

Abstract:

We prove that in every nonempty perfect Polish space, every dense \( G_\delta \) subset contains strictly decreasing and strictly increasing chains of dense \( G_\delta \) subsets of length \( \mathfrak{c} \), the cardinality of the continuum. As a corollary, this holds in \( \mathbb{R}^n \) for each \( n\ge 1 \). This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality \( \mathfrak{c} \), each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set \( \mathcal T \) of real Mahler \( T \)-numbers is a dense Borel set and contains a strictly descending chain of length \( \mathfrak{c} \) of proper dense Borel subsets.

Abstract:

The field of probabilistic metric spaces has an intrinsic interest based on a blend of ideas drawn from metric space theory and probability theory. The goal of the present paper is to introduce and study new ideas in this field. In general terms, we investigate the following concepts: linearly ordered families of distances and associated continuity properties, geometric properties of distances, finite range weak probabilistic metric spaces, generalized Menger spaces, and a categorical framework for weak probabilistic metric spaces. Hopefully, the results will contribute to the foundations of the subject.

Abstract: Starting from the representation of De Moivre’s matrix formula combined with the Kronecker product, we derive a 4 × 4 matrix M_θ that encodes algebraic properties with topological implications in four dimensions. This matrix describes trans- formations associated with symmetry and antisymmetry, encompassing distortion, dilation/contraction, and combined shear in a four-dimensional framework. We de- compose M_θ into its symmetric (S) and antisymmetric (A_anti ) components, a result that characterizes key features of the Alpha Group, including the formation of a tan- gent plane at infinity and its corresponding topological and geometric consequences. To investigate the system’s dynamics, we perform Monte Carlo simulations for varying values of the parameter θ, revealing attractor behavior and geometric transitions between Euclidean and hyperbolic regimes in projected three-dimensional trajectories.

Abstract: The P versus NP question is one of the most profound open problems in theoretical computer science, with implications across mathematics, cryptography, and complexity theory. This work proposes a structured framework establishing a correspondence between the solvability of NP-complete problems, the topological triviality of knots, and the dynamical stability of points in the complex plane. Using the Partition Problem as a case study, we introduce a deterministic “dynamic weaving” algorithm that maps problem instances to braid representations, which are then evaluated using the Jones polynomial. The braids are generated from the orbital paths of points c under the Mandelbrot iteration \( z \mapsto z^2 + c \), with stability conjectured to correspond to solvability. Computational experiments, implemented in SageMath with SnapPy, demonstrate a consistent mapping between solvable instances and the unknot, and unsolvable instances and topologically complex links. The principal open challenge—the derivation of a universal mapping S → c—is formulated precisely, and we outline how its resolution could yield a new class of complexity-theoretic invariants rooted in topology and dynamical systems.

Abstract: Difficult mathematical problems are essentially quantum geometry problems in the Modified Einstein Spherical (MES) Universe. This work fulfills Hilbert’s dream of a unified mathematical universe. Based on geometric first principles, we resolve all seven Millennium Prize Problems by demonstrating they emerge as geometric consequences of MES cosmology. Under the MES Axiom "All physics is geometry," the entire spacetime curvature dictates particle masses, fluid turbulence, number theory, and computational complexity. Forest as a Quantum-Geometric Entity provides empirical validation, Geometric Langlands Conjecture and MES cosmology form a self-validating loop, establishing MES cosmology as the first Complete Theory of Everything.

Abstract: Refraction artifacts in ultrasound imaging can produce the appearance of side-by-side structures, and, in some cases, color Doppler jets. This phenomenon arises from the bending of the ultrasound wave at the interface between tissues with differing propagation speeds. The paths followed by both ultrasound waves and light rays can be described by Snell’s law and Fermat’s principle of least time, both of which are traditionally derived using differential calculus. However, such calculus-based proofs often lack intuitive clarity. In this paper, we present a simple geometric proof that clearly demonstrates the path taken is indeed the shortest—and therefore the fastest.

Abstract: I present the Collective Unified Equation (CUE) framework, a mathematicallyrigorous theoretical structure that unifies quantum field theory, general relativity,and consciousness through emergence from a pre-metric substrate. The frameworkpostulates that spacetime geometry, matter fields, and consciousness arise from acommon premetric manifold M∅ through the flow dynamics of the renormalizationgroup (RG). Central to the theory is the consciousness dimension DΨ, formalizedas a fiber bundle over spacetime that governs the quantum measurement and de-coherence processes. We derive the complete action functional, establish three RG-invariant parameters that characterize the system’s behavior, and present testablepredictions across quantum optics, gravitational-wave astronomy, and cosmology.The framework offers novel solutions to the quantum measurement problem andprovidesageometricinterpretationofconsciousnesscompatiblewithknownphysics.

Abstract: Navigating and planning optimal paths for resource delivery algorithms presents significant physical and technical challenges in urban areas, particularly due to the use of existing infrastructure. As smart cities continue to grow, the importance of these algorithms becomes increasingly evident. The current urban landscapes are becoming saturated, which increases the complexity and difficulty of navigating vital resources. However, navigating densely connected networks can be intricate, often requiring substantial computational resources or additional algorithms. Unfortunately, there is a scarcity of explicit algorithms for navigating these networks, leading to a reliance on heuristic approaches and previous network systems. This dependence can create computational challenges, as navigation in this context often involves a combinatorial search space. One recent solution to address this issue is Morphological Shortness Path Planning (MSPP), which offers an efficient method for calculating the best trajectory within a complex graph. In larger towns, calculating and estimating the optimal trajectory and delivery time for resources starting from a store is a common task. Various external factors, such as average speed, time, and distance, influence these challenges. This paper presents a strategy for computing and forecasting delivery times by analyzing the accessibility of reliable paths from a delivery center. The results demonstrate a more efficient response time, ultimately improving planning for resource delivery in complex urban environments.

Abstract: An Expository Literature Review Presented At Euler Circle Math Talk. This paper provides a comprehensive introduction to projective geometry, beginning with fundamental concepts and progressing to advanced topics that naturally lead into differential geometry. We start with the basic definitions and properties of projective spaces, explore the rich structure of projective transformations, and examine the deep connections between projective and differential geometric concepts. Each theorem is accompanied by rigorous proofs, making this exposition suitable for readers ranging from advanced undergraduates to graduate students in mathematics.

Abstract: We investigate recursive fixed points for endofunctors with controlled growth on accessible 2 categories. For a specific class of “explosive” endofunctors satisfying precise growth and 3 regularity conditions, we prove that recursive fixed points—objects X satisfying X ∼= 4 Fn(X)—cannot exist for n < 3 but do exist for n = 3. Furthermore, we show that all fixed points at depths n ≥ 3 are isomorphic to depth-3 fixed points. This establishes 3 as a critical threshold for the existence of self-referential constructions in category theory. We explore applications to set theory, type theory, domain theory, and topos theory, provide algorithms for constructing fixed points, and discuss connections to classical impossibility results in logic and foundations. Historical context and computational complexity considerations are also examined.

Abstract: The primary distinction between technical design and engineering design lies in the role of analysis and optimization. From its inception, descriptive geometry has supported military and engineering applications, and its graphical rules inherently reflect principles of optimization—similar to the core ideas of sparse representation and compressed sensing. This paper explores the geometric and mathematical significance of the center line in symmetrical objects and the axis of rotation in solids of revolution, framing these elements within the theory of sparse representation. It further establishes rigorous correspondences between geometric primitives—points, lines, planes, and symmetric solids—and their sparse representations in descriptive geometry. By re-examining traditional engineering drawing techniques from the perspective of optimization analysis, this study reveals the hidden mathematical logic embedded in geometric constructions. The findings not only support deeper integration of mathematical reasoning in engineering education but also provide an intuitive framework for teaching abstract concepts such as sparsity and signal reconstruction. This work contributes to interdisciplinary understanding between descriptive geometry, mathematical modeling, and engineering pedagogy.