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Why is $e^{\pi \sqrt{163}}$ almost an integer?

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$e\pi \sqrt{163}=262537412640768743.99999999999925$ is very close to an integer. Coincidence? Not at all. In this talk, we will see how this is related to the $j$-function, a certain holomorphic function on the upper half plane. We will interpret the $j$-function as a function on the space of elliptic curves, and then see how symmetries of elliptic curves and of this space have far-reaching consequences—not only for $e\pi \sqrt{163}$.

This talk is part of the Trinity Mathematical Society series.

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