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Explore hyperbolic localization techniques in Donaldson-Thomas theory through this advanced mathematics lecture delivered at Kansas State University's M-Seminar. Delve into the hyperbolic localization functor introduced by Braden, which restricts constructible complexes from schemes with one-dimensional torus actions to attracting varieties and projects with compact support to fixed varieties, extending Białynicki-Birula decompositions beyond smooth cases. Examine Richarz's proof that this functor commutes with vanishing cycles and discover 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 DT perverse sheaf construction by Brav, Bussi, Dupont, Joyce, and Szendroï, which produces cohomological DT invariants by gluing vanishing cycles on local models with careful treatment of quadratic form actions using orientation data. Understand the derivation of formulas for hyperbolic localization of DT perverse sheaves by combining Białynicki-Birula and Richarz results with analysis of hyperbolic localization behavior under quadratic forms and orientations, leading to critical versions of Białynicki-Birula decomposition in cohomological DT theory. Discover the stacky generalization replacing attracting and fixed varieties with stacks of filtered and graded points, which enables proofs of fundamental DT theory results including the Kontsevich-Soibelman wall-crossing formula and construction of cohomological Hall algebras for CY3 categories.
