Hyperbolic Localization in Donaldson-Thomas Theory - Part 3 of 3

Explore advanced techniques in Donaldson-Thomas theory through this mathematical seminar lecture focusing on hyperbolic localization functors and their applications to moduli spaces of sheaves on Calabi-Yau threefolds. Learn how the hyperbolic localization functor restricts constructible complexes from schemes with torus actions to attracting varieties and projects to fixed varieties, building on Braden's foundational work and Richarz's theorem on commutation with vanishing cycles. Discover how shifted symplectic geometry and shifted Darboux theorems describe moduli spaces locally through critical loci of functions on smooth spaces, and understand the construction of DT perverse sheaves whose cohomology yields cohomological DT invariants. Examine the proof techniques for formulas describing hyperbolic localization of DT perverse sheaves, combining Białynicki-Birula and Richarz results with analysis of hyperbolic localization behavior under quadratic forms and orientations. Investigate the critical version of Białynicki-Birula decomposition in cohomological DT theory and explore stacky generalizations that replace standard varieties with stacks of filtered and graded points. Understand how these theoretical developments have enabled proofs of fundamental results including the Kontsevich-Soibelman wall-crossing formula and the construction of cohomological Hall algebras for CY3 categories, representing significant advances in modern algebraic geometry and mathematical physics.