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Explore the fundamental question of whether elliptic curve ranks can be arbitrarily large in this 56-minute conference talk from the Simons Foundation. Delve into the intersection of elliptic curves across multiple mathematical areas and their central role in modern number theory, focusing on how the rank of an elliptic curve over rational numbers measures the size of its group of rational points. Learn about computational approaches and statistical models used to investigate this unresolved question dating back to Poincaré, examining data patterns, heuristic frameworks, and challenging outliers that test current assumptions. Discover how the rank intuitively counts the number of independent points needed to generate all rational solutions up to torsion, and gain insights from collaborative research combining computational algebra with arithmetic geometry. Understand the broader implications for the Langlands program and how algorithmic approaches make abstract mathematical objects like modular forms, elliptic curves, Galois representations, and L-functions computationally accessible for large-scale experimental number theory.
