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Let $ \mathfrak{M}$ be the moduli space of torsion free $ G_2$ structures on a compact 7-manifold $ M$, and let $ \mathfrak{M}_1 \subset \mathfrak{M}$ be the $ G_2$ structures with volume($M$) $=1$. The cohomology map $ \pi^3: \mathfrak{M}…

We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at…

Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $\rho: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are…

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We initiate the study of flux compactifications on non-supersymmetric Riemann-flat manifolds (RFM's) with Casimir energy. While curvature and other corrections are suppressed in RFM's, the inclusion of Casimir energies allows one to evade…

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We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M.

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We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation,…