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Let G be a semi-direct product with A Abelian and K compact. We characterize spread-out probability measures on G that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on G that we develop, and which is analogous to the well-known Beurling-Gelfand spectral radius formula. For spread-out probability measures on G , we also characterize ergodicity by convolutions by means of the Fourier transform of the measure. Finally, we show that spread-out probability measures on such groups are mixing by convolutions if and only if they are weakly mixing by convolutions.

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Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M . We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.

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We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori.

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Let Γ be a genus 0 group between Γ 0 ( N ) and its normalizer in SL 2 (ℝ) where N is squarefree. We construct an automorphic product on Γ × Γ and determine its sum expansions at the different cusps. We obtain many new product identities generalizing the classical product formula of the elliptic j -function due to Zagier, Borcherds and others. These results imply that the moonshine conjecture for Conway's group Co 0 is true for elements of squarefree level.

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By a theorem of McLean, the deformation space of an associative submanifold Y of an integrable G 2 manifold ( M, ϕ ) can be identified with the kernel of a Dirac operator on the normal bundle ν of Y . Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through ‘pseudo-associative’ submanifolds. Infinitesimally, this corresponds to twisting the Dirac operator with connections A of ν . Furthermore, the normal bundles of the associative submanifolds with Spin c structure have natural complex structures, which helps us to relate their deformations to Seiberg-Witten type equations. If we consider G 2 manifolds with 2-plane fields ( M , ϕ, λ) (they always exist) we can split the tangent space TM as a direct sum of an associative 3-plane bundle and a complex 4-plane bundle. This allows us to define (almost) λ-associative submanifolds of M , whose deformation equations, when perturbed, reduce to Seiberg-Witten equations, hence we can assign local invariants to these submanifolds. Using this we can assign an invariant to ( M , ϕ, λ). These Seiberg-Witten equations on the submanifolds are restrictions of global equations on M . We also discuss similar results for the Cayley submanifolds of a Spin(7) manifold.

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We prove that the essential range of the gradient of planar Lipschitz maps has a connected rank-one convex hull. As a corollary, in combination with the results in [ Faraco, D., and Székelyhidi, Jr., L. , Tartar's conjecture and localization of the quasiconvex hull in R 2x2 , Acta Math., to appear.] we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank-one convexity.