2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146963Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.Lecture notes for the Solomon Lefschetz Memorial Lecture Series, September 8-9, 2005 by Maxim Kontsevich at Cinvestav, Mexico. Notes by Ernesto Lupercio. http://www.math.cinvestav.mx/KontsevichEnglish.htm. To appear in Contemporary MathematicsAlgebraic GeometryMathematical PhysicsK-Theory and Homology14A22; 14G10text