Exponential sums is one of the central tools in number theory with endless number of application. The study of exponential sums combines tools from analysis and algebra. In fact, bounding exponential sums was one of the main motivations for the development of algebraic geometry. In particular, two of major results of the 20th century where motivated by bounding exponential sums: Weil's Riemann hypothesis for curves bounds one parameter sums, and Deilgne's resolution of the Weil's conjectures deals with exponential sums with several parameters.
The theory f exponential sums remains central and sees a lot of recent breakthroughs.
The course aims to provide an entry point for a student into this area, by covering central classical topics.
In particular, we will discuss: Gauss and Jacobi sums, Kloosterman sums, moments bounds, equidistribution results, equations over finite fields, diagonal hypersurfaces, Zeta functions of hypersurfaces, Riemann hypothesis for sums in one variable, and the Hasse-Davenport relation.