Character Sums and Their Applications

Abstract

Character sums play significant roles in analytic number theory. In particular, bounding additive and multiplicative characters of the integers modulo $q$ as well as of finite fields are classical questions that have led to many fruitful and far-reaching applications. In this thesis, we first provide new bounds on generalizations of multiplicative character sums in higher dimensions. Then we develop new applications, based on joint work with Chen An and Lillian Pierce, of additive character sums to questions in PDEs.