Kontsevich, Maxim XI Solomon Lefschetz memorial lecture series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio). (English) Zbl 1155.53060 Dito, Giuseppe (ed.) et al., Non-commutative geometry in mathematics and physics. The XI Solomon Lefschetz memorial lecture series and topics in deformation quantization and non-commutative structures, Mexico City, Mexico, September 7–9, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4147-1/pbk). Contemporary Mathematics 462, 1-21 (2008). This paper contains the shortened lecture notes of two lectures given by the author at the XI Solomon Lefschetz memorial lecture series and topics in deformation quantization and non-commutative structures [Mexico City, Mexico, September 7–9 (2005; Zbl 1141.58003)].He presents a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry. After presenting the Hodge structures and some elements of non-commutative geometry, he talks about the Hodge filtration on \(HP_*(A)\) and the main problems in non-commutative geometry. Then he presents the basic derived algebraic geometry and the operation with saturated spaces. Finally, he studies the degeneration conjecture as well as the non-commutative Hodge conjecture.For the entire collection see [Zbl 1141.58003]. Reviewer: Vasile Oproiu (Iaşi) Cited in 17 Documents MSC: 53D55 Deformation quantization, star products 58A14 Hodge theory in global analysis 14A22 Noncommutative algebraic geometry Keywords:Hodge structures; non-commutative geometry; Hodge filtration; saturated spaces; degeneration conjecture Citations:Zbl 1141.58003 PDFBibTeX XMLCite \textit{M. Kontsevich}, Contemp. Math. 462, 1--21 (2008; Zbl 1155.53060) Full Text: arXiv