Pacific Institute for the Mathematical Sciences - PIMS

Title: Numbers and Shapes

Abstract: 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. 

Bio:

Henri Darmon specializes in number theory, working on Hilbert's 12th problem and its relation with the Birch-Swinnerton-Dyer conjecture. He is a Fellow of the Royal Society of Canada and currently, a James McGill Professor of Mathematics at McGill University.

This is the second lecture in the Hugh C. Morris Lecture Series. See the media release
http://www.pims.math.ca/news/hugh-morris-former-chair-pims-board-endows-...

Location: University of Calgary, Room ICT 122