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Analytic Number Theory Graduate Student Seminar February 28, 2017 (04:00 PM PST - 05:00 PM PST)
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Location: MSRI: Baker Board Room
Speaker(s) Aled Walker (University of Oxford), Alexander Walker (Brown University)
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4.00pm: Aled Walker

Title: Gowers norms control Diophantine inequalities

Abstract: The Generalised von Neumann Theorem of Green and Tao is a central tool in the study of systems of linear equations with integer coefficients. In this talk we will discuss how this theorem may be adapted to tackle Diophantine inequalities (namely systems of linear inequalities with irrational coefficients), showing that Gowers norms control the number of solutions. Time permitting, we will consider the case of inequalities with prime variables. 


4.30pm: Alex Walker

Title: Sums of Fourier Coefficients of Modular Forms

Abstract: Several arithmetic problems can be encoded into questions about the growth of Fourier coefficients of modular forms.  The Gauss Circle Problem, for example, concerns bounding the error in partial sums of coefficients attached to powers of the theta function.  In this talk I will explain how these partial sums can be understood via Dirichlet series, an atypical approach in a field dominated by exponential sum methods.

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