# Noncommutative Hodge theory learning seminar

## Schedule

Organizer: Brent Pym

Winter 2018 (Semester 2)

Mondays and/or Fridays at 16:10

JCMB ~~5327~~ 6206

Date | Speaker | Topic | Notes (no guarantee of correctness) |

Jan 22 | Brent Pym | Introduction and motivation | nc-hodge-01.pdf |

Jan 29 | Matt Booth | Hochschild homology for algebras and dg categories | nc-hodge-02.pdf |

Feb 5 | Tim Weelinck | The Hochschild-Kostant-Rosenberg theorem | nc-hodge-03.pdf |

Feb 12 | David Jordan | Cyclic homology and the NC Hodge filtration | nc-hodge-04.pdf |

Mar 9 | Sjoerd Beentjes | The Gysin triangle | |

Mar 26 | Sue Sierra | Invariants of noncommutative projective schemes | |

Apr 9 | Theo Raedschelders | The Hodge-de Rham degeneration theorem I | |

Apr 13 | Theo Raedschelders | The Hodge-de Rham degeneration theorem II | |

Apr 23 | Peter Samuelson | K-theory and the Chern character | |

Apr 30 | Johan Martens | The 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
- ...

## References and links

### 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

### K-theory

Blanc, Topological K-theory of complex noncommutative spaces

Weibel, The K-book: an introduction to algebraic K-theory

### Riemann-Roch

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