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How loud is an arithmetic drum?

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A modular form can be viewed as a wave function on an arithmetic surface. The theory of quantum chaos asks how these waves distribute as their energy grows. One approach is the so-called sup-norm problem, where one looks for bounds on the size of automorphic forms as certain parameters grow, such as the weight or eigenvalue, and the level. This talk will be a gentle introduction to this rich problem, touching on spectral theory, the amplification method, algebraic number theory, and applications to L-functions.

This talk is part of the Junior Algebra and Number Theory seminar series.

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