Mathematical Sciences - Colloquium

Andrew Wiles' proof of Fermat's Last Theorem relies in turn on his proof that a wide class of elliptic curves are "modular" (this is the so-called Taniyama-Shimura conjecture). I will begin by describing what this conjecture says in concrete, friendly terms.

But elliptic curves are not the only objects thought to be modular. In recent work, Ken Ono and I have devised a new method which proves the modularity of a certain "Calabi-Yau threefold". As a result we can prove Beukers' conjectured "supercongruence" for the Apery numbers (these are combinatorial sums introduced by Apery to establish the irrationality of certain values of the Riemann zeta function).

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