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Explore advanced mathematical concepts in this graduate-level seminar lecture focusing on hyperbolic localization techniques within Donaldson-Thomas theory. Delve into the hyperbolic localization functor introduced by Braden, which restricts constructible complexes from schemes with torus actions to attracting varieties and projects them to fixed varieties, particularly useful for studying Białynicki-Birula decompositions in non-smooth cases. Examine Richarz's fundamental result demonstrating that this functor commutes with vanishing cycles, and understand how shifted symplectic geometry and shifted Darboux theorems describe moduli spaces of sheaves on Calabi-Yau threefolds through critical loci of functions on smooth spaces. Learn about the construction of DT perverse sheaves by Brav, Bussi, Dupont, Joyce, and Szendroï, which involves gluing vanishing cycles on local models with careful attention to orientation data and quadratic form trivializations. Discover the proof techniques for establishing formulas for hyperbolic localization of DT perverse sheaves, combining Białynicki-Birula and Richarz results with analysis of hyperbolic localization behavior under quadratic forms and orientations. Understand how these methods yield critical versions of Białynicki-Birula decompositions in cohomological DT theory and explore the stacky generalizations that replace traditional varieties with stacks of filtered and graded points, leading to breakthrough results including proofs of the Kontsevich-Soibelman wall-crossing formula and construction of cohomological Hall algebras for CY3 categories.
