Alex Iosevich

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Venue:DM 409A

Abstract: The classical regular value theorem says that if f: X -> Y is an immersion, where X,Y are smooth manifolds of dimension n,m , n>m , respectively, then the set {x in X: f(x)=y } is either empty or is an n-m dimensional sub-manifold of X . We shall see that a suitable analog of this result is available if a manifold X is replaced by a set of sufficiently large Hausdorff dimension and the function f satisfies a "rotational curvature" condition. Regularity of generalized Radon transforms plays a key role. Sharpness results are based on an interplay between ideas from discrete geometry and number theory.