Kavli Affiliate: Pau Amaro Seoane
| Summary:
We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($mathbbH$ and $mathbbO$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $mathbbH$, $G_2$ for $mathbbO$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($Δ=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $mathcalV(x) = |P(x)|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_rm collapse propto ε^-2$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.
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