Wooley, Trevor D. Diophantine methods for exponential sums, and exponential sums for Diophantine problems. (English) Zbl 1022.11051 Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 207-217 (2002). Summary: Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal Diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.For the entire collection see [Zbl 0993.00022]. MSC: 11P55 Applications of the Hardy-Littlewood method 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11L07 Estimates on exponential sums 11P05 Waring’s problem and variants 11D72 Diophantine equations in many variables 14G05 Rational points Keywords:Hardy-Littlewood method; exponential sums; Waring’s problem; equations in many variables; rational points; representation problems PDFBibTeX XMLCite \textit{T. D. Wooley}, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 207--217 (2002; Zbl 1022.11051) Full Text: arXiv