| 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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