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Abstract: The naive translation of the Torelli theorem from K3 surfaces to
Calabi-Yau threefolds does not work, and counterexamples to it are helpful
for understanding what the correct statement should be. In the first part of
my talk I shall attempt to explain how the question of finding
counterexamples to Torelli is related to equivalences of derived categories.
Then, I'll atempt to construct such a counterexample using the
Ogg-Shafarevich theory for elliptic threefolds. The equivalence of derived
categories I construct is best explained through the use of the Jacobian
fibration and twisted sheaves. If time will allow I shall attempt to explain
why singularities appear and their relationship to the stable singularities
in Mirror Symmetry, as well as how this relates to questions from algebra,
like derived equivalences for Azumaya algebras and tilting modules.
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