Noncommutative Hodge theory learning seminar


Organizer: Brent Pym
Winter 2018 (Semester 2)
Mondays and/or Fridays at 16:10
JCMB 5327 6206

DateSpeakerTopicNotes (no guarantee of correctness)
Jan 22Brent PymIntroduction and motivationnc-hodge-01.pdf
Jan 29Matt BoothHochschild homology for algebras and dg categoriesnc-hodge-02.pdf
Feb 5Tim WeelinckThe Hochschild-Kostant-Rosenberg theoremnc-hodge-03.pdf
Feb 12David JordanCyclic homology and the NC Hodge filtrationnc-hodge-04.pdf
Mar 9Sjoerd BeentjesThe Gysin triangle
Mar 26Sue SierraInvariants of noncommutative projective schemes
Apr 9Theo RaedscheldersThe Hodge-de Rham degeneration theorem I
Apr 13Theo RaedscheldersThe Hodge-de Rham degeneration theorem II
Apr 23Peter SamuelsonK-theory and the Chern character
Apr 30Johan MartensThe Gauss-Manin connection
... ... ... ...

Possible future topics, in no particular order:

  • Topological K-theory
  • Kontsevich-Katzarkov-Pantev definition of an NC Hodge structure
  • Hodge structures of deformation quantization algebras via cyclic formality
  • Examples from matrix factorizations
  • Examples from symplectic topology and mirror symmetry
  • ...

Lecture notes

Kaledin, Lecture notes from a mini-course in Tokyo in 2008
Stern et al, Notes from a similar seminar in Bonn in 2016

Classical Hodge theory

Deligne, Théorie de Hodge II and III
Filippini-Ruddat-Thompson, An introduction to Hodge structures
Griffiths-Harris, Principles of Algebraic Geometry
Peters-Steenbrink, Mixed Hodge structures
Voisin, Hodge Theory and Complex Algebraic Geometry I

Noncommutative Hodge structures

Kontsevich, Solomon Lefschetz Memorial Lectures on "Hodge structures in non-commutative geometry"
Katzarkov-Kontsevich-Pantev, Hodge theoretic aspects of mirror symmetry
Sabbah, Introduction to pure non-commutative Hodge structures

Noncommutative motives

Tabuada, Higher K-theory via universal invariants
Tabuada, Noncommutative motives

Differential graded categories

Drinfeld, DG quotients of DG categories
Keller, On differential graded categories
Thomason-Trobaugh Higher Algebraic K-Theory of Schemes and of Derived Categories
Toën, Lectures on DG-categories

Cyclic homology

Connes, Non-commutative differential geometry
Loday, Cyclic homology
Kassel, Cyclic homology, comodules, and mixed complexes
Keller, Invariance and localization for cyclic homology of DG algebras
Keller, On the cyclic homology of exact categories
Keller, On the Cyclic Homology of Ringed Spaces and Schemes
Tsygan, Cyclic homology
Voigt, Introduction to cyclic homology
Weibel, Cyclic homology for schemes
Weibel, An introduction to homological algebra

Gauss-Manin connection

Dolgushev-Tamarkin-Tsygan, Noncommutative calculus and the Gauss-Manin connection
Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology
Tsygan, On the Gauss-Manin connection in cyclic homology
Sheridan, Formulae in noncommutative Hodge theory

Hodge to de Rham degeneration

Kontsevich-Soibelman, Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry (statement of the conjecture)
Kaledin, several papers on the proof math/0511665, math/0611623, 0708.1574, 1601.00637
Mathew, Kaledin's degeneration theorem and topological Hochschild homology


Blanc, Topological K-theory of complex noncommutative spaces
Weibel, The K-book: an introduction to algebraic K-theory


Shklyarov, Hirzeburch-Riemann-Roch theorem for differential graded algebras

Deformation quantization

Dolgushev, A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold
Dolgushev-Tamarkin-Tsygan, Formality theorems for Hochschild complexes and their applications
Kontsevich, Deformation quantization of Poisson manifolds and algebraic varieties
Shoikhet, A proof of the Tsygan formality conjecture for chains
Willwacher, Formality of cyclic chains

Matrix factorizations/Landau-Ginzburg models

Shklyarov, Non-commutative Hodge structures: Towards matching categorical and geometric examples
Dyckerhoff, Compact generators in categories of matrix factorizations
Katzarkov-Kontsevich-Pantev, Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models

Hodge theory of generalized complex manifolds

Cavalcanti, The decomposition of forms and cohomology of generalized complex manifolds
Gualtieri, Generalized geometry and the Hodge decomposition
Gualtieri, Generalized complex manifolds

Videos of talks

Pantev, Hodge Structures in Symplectic Geometry
Kontsevich, Noncommutative Motives
Shklyarov, Semi-infinite Hodge structures in noncommutative geometry