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Mathematics Physics

10

Open Unknowns

0

Cross-Domain Bridges

10

Active Hypotheses

Open Unknowns (10)

Unknown Is Almgren's regularity theorem — that the singular set of an area-minimising current in ℝⁿ has Hausdorff dimension ≤ n-2 (and ≤ n-8 for hypersurfaces) — sharp, and what is the correct singular set dimension for physically relevant minimal surfaces in curved Riemannian manifolds? u-almgren-regularity-singular-set-sharp-dimension

Unknown Are Thom's seven elementary catastrophes truly complete for codimension ≤4, and do higher-codimension catastrophes appear in real physical or biological systems? u-catastrophe-normal-form-completeness

Unknown How does classical chaos emerge from quantum mechanics in the semiclassical limit — and is there a quantum analog of the classical Lyapunov exponent that characterizes information scrambling in many-body quantum systems? u-chaos-quantum-correspondence-lyapunov-exponents

Unknown Under what conditions do quantum many-body systems fail to thermalise (violate ergodicity) and what is the complete classification of ergodic vs. non-ergodic phases? u-ergodic-failure-quantum-thermalization

Unknown Whether the full spectrum of pseudo-Goldstone bosons in realistic symmetry-breaking scenarios (explicit breaking, finite volume, non-equilibrium) can be systematically classified beyond the idealised Goldstone theorem, and what their masses imply for phase transitions in novel materials u-goldstone-boson-higher-dimensional-systems

Unknown How do integrability-breaking perturbations (higher-order dispersion, dissipation, noise, higher-dimensional geometry) degrade soliton stability and modify spreading speed in realistic physical systems — and is there a general perturbation theory for nearly-integrable systems that predicts the soliton lifetime and effective dynamics? u-integrability-breaking-perturbations-soliton-stability-realistic-systems

Unknown Does quantum probability admit a rigorous measure-theoretic foundation that accounts for contextuality and non-commutativity — and if so, what is the correct σ-algebra over which quantum events are defined? u-measure-theoretic-foundations-quantum-probability

Unknown Is the exact (Wilsonian) renormalization group complete — does it capture all universality classes and phase transitions including those with no perturbative expansion? u-nonperturbative-rg-completeness

Unknown In interdependent networks (e.g. power grid ↔ internet), where failure of nodes in one network triggers failures in the other, the percolation transition becomes first-order (abrupt) — what determines the critical coupling strength between network layers below which the transition reverts to second-order, and can this threshold be engineered to prevent catastrophic infrastructure cascades? u-percolation-phase-transition-interdependent-networks-cascading-failures

Unknown What is the precise mathematical relationship between symplectic geometry and quantum mechanics beyond deformation quantization, and when does geometric quantization fail to produce physically correct Hilbert spaces? u-symplectic-quantization-gap

Active Hypotheses

Hypothesis The cusp catastrophe control surface is topologically equivalent to the Landau free energy surface for all mean-field first-order transitions, predicting that hysteresis loops and spinodal boundaries are universal across physical, chemical, and biological systems sharing the same order-parameter symmetry class. high

Hypothesis Quantum mechanics is the deformation quantization of classical symplectic mechanics: the non-commutative algebra of quantum observables (A*B - B*A = iħ{A,B}_Poisson) is a formal deformation of the commutative Poisson algebra on phase space, with the symplectic structure ω providing the bracket, and symplectic integrators preserving ω correspond exactly to unitary quantum time evolution. high

Hypothesis The Feigenbaum universality of period-doubling routes to chaos (δ ≈ 4.669, α ≈ 2.502) extends to quantum maps via the quantum-classical correspondence: quantized versions of the logistic map and the Hénon map exhibit the same universal period-doubling ratios in the semiclassical limit (ℏ → 0, N_eff → ∞), with quantum corrections suppressed as O(ℏ) relative to classical universal behavior. medium

Hypothesis The KAM theorem's mechanism — preserved invariant tori in near-integrable systems — has a quantum analogue in many-body localisation, where disorder preserves an extensive number of approximate local integrals of motion (LIOMs) that prevent thermalisation, making MBL the quantum KAM theorem. high

Hypothesis Gleason's theorem establishes that the Born rule is the unique probability measure on the Hilbert space lattice of projections, implying that the probabilistic structure of quantum mechanics is not an additional postulate but a mathematical consequence of the Hilbert space formalism for dimensions ≥ 3. medium

Hypothesis The quantum inverse scattering method (Faddeev-Takhtajan-Sklyanin, 1978-1982) is the exact quantum analog of the classical inverse scattering transform, with the quantum R-matrix playing the role of the classical Lax pair, and the Bethe ansatz eigenvalues corresponding to quantized soliton momenta — unifying classical and quantum integrability into a single algebraic framework (Yangians). high

Hypothesis Deep neural networks at the edge of chaos (criticality) exhibit renormalization group fixed-point behavior: representations across network layers correspond to RG flow toward an IR fixed point, and the universality class of the fixed point determines generalization capacity independent of architecture details. medium

Hypothesis Every continuous phase transition in nature corresponds to the spontaneous breaking of a specific symmetry group G → H, and the universality class (critical exponents) is determined entirely by the dimension of G/H and the spatial dimension d — making the Landau-Ginzburg-Wilson classification complete for all equilibrium phase transitions medium

Hypothesis Targeted vaccination of high-degree nodes (hubs) in scale-free contact networks achieves epidemic suppression (R_eff < 1) with the number of doses equal to O(√N) rather than the O(N·(1-1/R₀)) doses required by random mass vaccination, and this reduction is achievable in practice using acquaintance immunisation (vaccinate a random contact of a random person). high

Hypothesis Biological cell membrane shapes during morphogenesis are Willmore energy (W = ∫H² dA) minimizers subject to volume and area constraints, with active cytoskeletal forces entering as Lagrange multipliers that define the morphogenetic Willmore flow, and the shapes of cell organelles (ER, Golgi) are calculable from the Helfrich elastic model without free parameters. medium

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