Sum of Kloosterman Sums, Singular Exponential Sums and Their Applications

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The investigator studies the methods of estimating the sum of exponential sums, especially the sum of the Kloosterman sums, which explores the subtle cancellation of the terms and which goes beyond the bounds of Weil and Deligne for exponential sums. The applications of such methodsto two outstanding problems are highlighted. These sums are fundamental objects and arise naturally from a variety of open problems in number theory and harmonic analysis. The investigator studies the optimal way to estimate them, which potentially has strikingapplications to several well known unsolved problems.