Mathematical Physics

2606 Submissions

[2] viXra:2606.0052 [pdf] submitted on 2026-06-13 23:16:58

Towards a Floer Theory for Mars II Floer Hessian Field Almost Extends

Authors: Urs Frauenfelder, Joa Weber
Comments: 57 pages, 2 figures

In part I, [FW26a], we showed that collisional periodic orbits of twisted Zeeman systems can be detected variationally by a non-local Hamiltonian action functional. In this part II we show that the linearized gradient flow of this non-local functional is a Fredholm operator and prove a non-local elliptic regularity result. These results are obtained with the theory of almost extendability of weak Hessian fields introduced in [FW26c].
Category: Mathematical Physics

[1] viXra:2606.0005 [pdf] submitted on 2026-06-02 14:44:15

Geometric and Topological Approaches to Crystallography

Authors: Ellie Richwine, Lucian M. Ionescu
Comments: Pages.

This article explores the mathematical structures underpinning crystalline materials, bridging the gap between pure mathematics and materials science. Building upon Toshikazu Sunada’s breakthrough framework of topological crystallography and subsequent formalizations by John C. Baez, we provide a rigorous yet accessible introduction to the geometric and topological modeling of crystals. The study examines polyhedral geometry, duality, and lattice arrangements such as the Eisenstein and triangular lattices, framing them within the context of covering maps and Abel-Jacobi maps. Furthermore, we advance this foundation by introducing a simplified formulation of Graph Cohomology based on short exact sequences of graphs. This homological approach provides a unifying architectural template capable of tracking lattice defects via integer cohomology and modeling macroscopic continuous phenomena from discrete microscopic networks. The paper concludes by discussing the broader applications of these tools in molecular biology, theoretical physics, and fault-tolerant quantum engineering.
Category: Mathematical Physics