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Explore a groundbreaking mathematical lecture that presents a counterexample to Shioda's 1977 conjecture on unirationality in algebraic geometry. Delve into the fundamental differences between characteristic zero and positive characteristic in algebraic geometry, beginning with Castelnuovo's classical result that every unirational surface is rational in characteristic zero. Examine how this breaks down dramatically in positive characteristic, where non-rational and even general-type surfaces can be unirational. Learn about Shioda's proposed explanation through Galois representations, which conjectured that a simply-connected surface is unirational if and only if it is supersingular. Discover the speaker's construction of a counterexample that refutes this long-standing conjecture, involving a novel obstruction technique inspired by hyperbolicity studies in complex varieties. Gain insights into advanced topics in algebraic geometry including unirational and rational surfaces, supersingular varieties, Galois representations, and connections to hyperbolic geometry, presented as part of the Joint IAS/Princeton University Number Theory seminar series.
