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Explore a specialized mathematical lecture examining the Thomas-Yau conjecture through the lens of toric geometry and symplectic structures. Delve into the intricate relationship between Lagrangian sections contained within Calabi-Yau Lagrangian fibrations, specifically those that serve as mirrors of toric weak Fano manifolds. Learn how the speaker proves that a particular form of the Thomas-Yau conjecture holds in this toric case, demonstrating that a Lagrangian section L is Hamiltonian isotopic to a special Lagrangian section if and only if a specific stability condition is satisfied. Understand the mathematical framework involving slope inequalities on objects within exact triangles in the Fukaya-Seidel category, which aligns with general proposals by Li. Discover how, under more restrictive assumptions, these results establish a precise connection with Bridgeland stability, confirming predictions made by Joyce. Gain insights into cutting-edge research in symplectic geometry, algebraic geometry, and mathematical physics, as presented by a researcher from SISSA, Trieste, as part of the collaborative IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar series.
