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Mathematics Colloquium


Date:  Tuesday, April 19, 2011

Title:  Quantitative equidistributions of nilflows and Weyl sums

Abstract:  Bounds on Weyl sums (exponential sums of polynomial sequences) are an important classical topic in analytic number theory with at least a century of history (Hardy and Littlewood, Weyl, Vinogradov, Hua, Vaughan, Wooley and many others). In our joint work with L. Flaminio, following Furstenberg we approach the problem as a question on the equidistribution of nilflows (or linear skew-shifts). Our results are not far behind the best results obtained only very recently by number theoretical methods (Wooley, 2011) and we hope that our methods may yield further progress. Our work is based on ideas from the theory of dynamical systems such as finding solutions of cohomological equations, invariant distributions and renormalization, and it is part of a more general program to develop a theory of weakly chaotic, parabolic systems. Interval exchange transformations or translation flows, horocycle flows and nilflows are the main examples and our work is in fact an attempt at generalizing the Kontsevich-Zorich picture for the deviation of ergodic averages of interval exchange transformations.

Speaker:  Giovanni Forni
Institution:  Univ of Maryland


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