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MA939 - Topics in Number Theory

Lecturer: Sam Chow

Term(s): Term 2

Commitment: 30 lectures

Timetable: Tu 10-11 (B1.01), Wed 13-14 (D1.07) and Th 12-13 (D1.07)

Assessment: Oral exam

Prerequisites: Assumed: undergraduate-level algebra, analysis, number theory. Useful: Fourier analysis, complex analysis, local fields

Content : The Hardy–Littlewood circle method is a versatile approach to counting solutions to systems of diophantine equations. It uses Fourier analysis, and is effective when the number of variables is large compared to the degree. The plan is to cover the following material at varying depths: Waring's problem, Vinogradov's mean value theorem, the Davenport–Heilbronn method, Green's Fourier-analytic transference principle.

References: H. Davenport (2005), Analytic methods for Diophantine equations and Diophantine inequalities.
R. C. Vaughan (1997), The Hardy–Littlewood method.
S. J. Miller and R. Takloo-Bighash (2006), An invitation to modern number theory.
M. B. Nathanson (1996), Additive number theory: the classical bases.