# Bigger is Better

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Kevin Hughes
Fine Hall 314

Originally, Hardy and Littlewood developed their "circle method" to study Waring's problem on the representation of numbers as the sums of $k^th-powers$. In the circle method, one decomposes the circle into "major" and "minor" arcs. Some rough estimates on the minor arcs give a power saving, and the work is then to study the major arcs. The guiding principle is "Bigger is better", i.e. the best estimates arise from making the major arcs as large as possible. Recently, the circle method has been applied to discrete analogues in harmonic analysis. I will discuss the classical circle method, the spherical maximal function and higher degree analogues, and then discuss how these combine to give a discrete spherical maximal function.