Quantum Geometry, Topological Phases, and New Mathematical Structures: An Interdisciplinary Perspective

Quantum Geometry, Topological Phases, and New Mathematical Structures: An Interdisciplinary Perspective Mehmet Keçeci Orcid: https://orcid.org/0000-0001-9937-9839 Received: 2025-05-20Accepted: 2026-11-03Published: 2026-11-03 Abstract The rapid advancements in quantum technologies over the past two decades have necessitated not only new physical systems but also a fundamental transformation of the mathematical frameworks used to describe them. Hilbert spaces [1], topological phases of matter [2], nodal-line semimetals [2], and Weyl semimetals [3] represent paradigm shifts that bridge condensed matter physics, quantum information science, and functional analysis. In parallel, newly proposed mathematical constructs—including Keçeci Numbers, Keçeci Fractals, and Keçeci Layouts [4–9]—offer novel combinatorial and geometric tools that can sharpen the modeling of these physical structures. This article adopts a deliberately interdisciplinary approach, examining how quantum mechanics, topological phases of matter, and emerging mathematical frameworks can be unified within a coherent theoretical narrative. We focus on three axes: (i) the transition regions between classical and quantum physics, (ii) computational applications and their geometric underpinnings, and (iii) topological and geometric analyses that reveal hidden symmetries in both physical and mathematical systems. We further discuss the implications of the Second Quantum Revolution for nanomaterial-based quantum devices, the role of higher-derivative instanton solutions in the Keçeci–Fujii framework, and the potential of Keçeci combinatorial structures in quantum algorithm design. Keywords: Hilbert Space, Keçeci Numbers, Keçeci Fractals, Keçeci Layout, Quantum Geometry, Topology, Topological Phases, Quantum Computing, Qubit, Superposition, Entanglement, Unitary Dynamics, Quantum Mechanics, Functional Analysis, Operator Theory, Dirac Notation, Quantum States, Linear Operators, Quantum Algorithms, Quantum Information, Nodal-Line Semimetal, Weyl Semimetal.