Related papers: Algebraic Phase Theory I: Radical Phase Geometry a…
Algebraic Phase Theory (APT) exhibits a marked structural selectivity. In certain mathematical and physical settings it gives rise to rigidity phenomena, constrained representation behaviour, and reductions in apparent degrees of freedom,…
We study finite-depth reconstruction frameworks based on representation theory and show that non-rigid reconstruction behaviour is naturally accompanied by intrinsic structural boundaries. Within the finite-depth setting considered in this…
We develop the representation theory intrinsic to Algebraic Phase Theory (APT) in regimes where defect and canonical filtration admit faithful algebraic realisation. This extends the framework introduced in earlier work by incorporating a…
We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms,…
We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These…
We describe the development of Analytic Perturbation Theory (APT) in QCD, called Fractional APT (FAPT), which has been suggested to apply the renormalization group evolution and QCD factorization technique in the framework of APT.
The subject of the first section-lecture is concerned with the strength and the weakness of the perturbation theory (PT) approach, that is expansion in powers of a small parameter $\alpha$, in Quantum Theory. We start with outlining a…
The purpose of this article is to discuss recent advances in the growing field of phase retrieval, and to publicize open problems that we believe will be of interest to mathematicians in general, and algebraists in particular.
We introduce a perturbative approach to solving the time dependent Schroedinger equation, named adiabatic perturbation theory (APT), whose zeroth order term is the quantum adiabatic approximation. The small parameter in the power series…
We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative…
We study canonical filtrations of finite-dimensional associative algebras and Lie algebras. These filtrations are defined via optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT), and appear to be a…
We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
Topology in quantum matter is typically associated with gapped phases. For example, in symmetry protected topological (SPT) phases, the bulk energy gap localizes edge modes near the boundary. In this work we identify a new mechanism that…
We investigate the rupture of invariant tubes in a class of nonautonomous dynamical systems arising from time-dependent Ermakov-type equations. Starting from an exactly tube-integrable reference system, we analyze a time-dependent invariant…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
The Phase Tensor (PT) marked a breakthrough in understanding and analysis of electric galvanic distortion but does not contain any impedance amplitude information and therefore cannot quantify resistivity without complementary data. We…
We investigate the dynamical and structural properties of absorbing phase transitions (APTs) within granular systems. Specifically, we examine a model for vibrofluidized systems of spherical grains, which undergo a transition from a state…
We construct in the K matrix formalism concrete examples of symmetry enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and non-chiral topological phases and demonstrate by…
Recursive algebraic data types (term algebras, ADTs) are one of the most well-studied theories in logic, and find application in contexts including functional programming, modelling languages, proof assistants, and verification. At this…