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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...
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...
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...
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...
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...
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...
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