# Numbers and Shapes

Number theory is concerned with Diophantine equations and their solutions, encoded in discrete structures involving integers, rational numbers or algebraic quantities. Topology studies the properties of shapes that are unchanged under continuous or smooth deformations, a technique of choice being the construction of appropriate homological invariants. It turns out--perhaps surprisingly to the uninitiated--that these invariants can be endowed with sufficient structure to capture a tremendous amount of arithmetic information. The powerful interplay between arithmetic and topological ideas underlies the most important breakthroughs in the study of Diophantine equations, such as Faltings’ proof of the Mordell Conjecture and Wiles’ proof of Fermat’s Last Theorem. It is also at the heart of more recent and still very fragmentary attempts to construct algebraic points on elliptic curves when their existence is predicted by the Birch and Swinnerton-Dyer conjecture. This lecture will attempt to give a non-technical sampler of some of the rich, fascinating interactions between arithmetic questions and topological insights.

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