Fix an analytic simply connected bounded domain Ω⊂R2. A smooth function ψ:Ω→R that is constant on ∂Ω is said to define a steady Euler solution on Ω if the Poisson bracket between ψ and its Laplacian vanishes. In other words, ψ is a steady Euler solution if {ψ,Δψ}=0.⋆ It is well known that a special class of solutions to (⋆) are those for which we can write Δψ=F(ψ)⋆⋆ for some F:R→R. We show here that every nonradial analytic steady Euler solution solves an equation of the type (⋆⋆). To prove this, we first establish some general facts about the relation between pairs of analytic functions (f,g) satisfying {f,g}=0. We then use some properties of the zero sets of analytic functions and the moving plane method to establish rigidity results about solutions to (⋆⋆) with overdetermined conditions.
We study two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second (almost multilinear representability) deals with approximate subspace arrangements. In both cases, we prove that determining whether an input matroid has such a representation is undecidable. Consequently, the conditional independence implication problem is also undecidable, providing an independent answer to a question posed by Geiger and Pearl, recently resolved by Cheuk Ting Li. These problems are also closely related to characterizing achievable rates in network coding and constructing secret sharing schemes. For example, another corollary of our work is that deciding whether an access structure admits an ideal secret sharing scheme is undecidable. Our approach reduces undecidable problems from group theory to matroid representation problems. Specifically, we reduce the uniform word problem for finite groups to entropic representability and the word problem for sofic groups to almost multilinear representability. A key part of this reduction involves modifying group presentations into forms where linear representations are generic in an appropriate sense when restricted to the generating set.
On a class of dynamical space-times which are asymptotic as t-OO to a stationary space-time containing a horizon 3e0, we show the existence of a unique null hypersurface 3e which is asymptotic to 3e0. This is a special case of a general unstable manifold theorem for perturbations of flows which translate in time and have a normal sink at an invariant manifold in space. Examples of horizons 3e0 to which our result applies include event horizons of subextremal Kerr and Kerr-Newman black holes as well as event and cosmological horizons of subextremal Kerr-Newman-de Sitter black holes. In the Kerr(-Newman) case, we show that 3e is equal to the boundary of the black hole region of the dynamical space-time.
We prove that area-minimizing submanifolds are not generically smooth, settling a conjecture of White that asks the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is max{d-5, d-c}, where d denotes the dimension of the submanifold and c denotes the codimension.
We provide a proof for the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) Feynman rule conjecture, which applies to all genus Gromov-Witten potentials of the quintic threefold. This is achieved by extracting an equivalent A-model Feynman rule from the N-Mixed-Spin-P field (NMSP) theory. For genus greater than 1, the Feynman rule effectively computes the genus g Gromov-Witten potential using 3g-3 initial data. As a result, the Yamaguchi-Yau equations and the genus 1 and 2 mirror formulas are confirmed. A notable discovery is that the propagators in the BCOV theory can be interpreted in the A-model as enumerative counts of chains of rational curves in the NMSP master space outside the quintic threefold. In this way these propagators afford direct visibility of BCOV Feynman structures in the moduli of NMSP fields. Furthermore, we demonstrate that the set of all propagators capable of yielding Feynman rules forms a torsor under a "gauge" group action, which includes all BCOV's "gauge"freedom of propagators.
