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The last few years have witnessed a number of developments in the arithmetic of elliptic curves, notably the proof that there are positive proportions of elliptic curves of rank zero and rank one for which the Birch—Swinnerton-Dyer conjecture is true. The proof of this landmark result relies on an appealing mix of diverse techniques arising from the newly resurgent field of arithmetic invariant theory, Iwasara theory, congruences between modular forms, and the theory of Heegner points and related Euler systems. The purpose of this workshop is to survey the proof of this theorem and to describe the new perspectives on the Birch—Swinnterton-Dyer conjecture which it opens up.

Invited speakers: Mirela Ciperiani (Texas, Austin), Ellen Eischen (Oregon), Benedict Gross (Harvard), Wei Ho (Michigan), Antonio Lei (Laval), Chao Li (Columbia), Kartik Prasanna (Michigan), Victor Rotger (Catalunya), Ye Tian ( Chinese Academy of Sciences), Eric Urban (Columbia), Rodolfo Venerucci (Duisberg-Essen), Xin Wan (Columbia), Xiaoheng Jerry Wang (Princeton), Andrew Wiles (Oxford), Wei Zhang (Columbia)