Lectures on Noncommutative Geometry

Abstract

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.

Preliminary draft, 124pp. Comments are welcome!

References in corpus (9)

• Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
• The non-commutative Weil algebra
• Quiver varieties and a non-commutative P^2
• Open Strings and Non-commutative Geometry of Branes on Group Manifolds
• The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism
• Noncommutative Geometry and Quiver algebras
• A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver
• Double derivations and Cyclic homology
• Moyal quantization of necklace Lie algebras

Cited by in corpus (38)

• Moduli of representations of quivers
• Mirror symmetry: from categories to curve counts
• On Serre duality for compact homologically smooth DG algebras
• Matrix factorisations and open topological string theory
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
• Hochschild (co)homology and Koszul duality
• Noncommutative Geometry and Quiver algebras
• On double Poisson structures on commutative algebras
• An effective criterion for Nielsen-Schreier varieties
• A higher limit approach to homology theories
• On a family of quivers related to the Gibbons-Hermsen system
• Relative critical loci and quiver moduli
• Remarks on the Formulation of Quantum Mechanics on Noncommutative Phase Spaces
• Projective-injective modules, Serre functors and symmetric algebras
• The derived contraction algebra
• An introduction to associative geometry with applications to integrable systems
• Representation homology of topological spaces
• Categorification of (induced) cell modules and the rough structure of generalized Verma modules
• Exact sequences, Hochschild cohomology, and the Lie module structure over the $M$-relative center
• Higher zigzag algebras
• Noncommutative deformation theory, the derived quotient, and DG singularity categories
• A Simple Construction of Derived Representation Schemes
• Noncommutative Geometry in the Framework of Differential Graded Categories
• Strong homotopy inner product of an A-infinity algebra
• Non-commutative moduli spaces of topological D-branes
• Spaces of Operad Structures
• Calogero-Moser Spaces over Algebraic Curves
• Non-commutative Courant algebroids and Quiver algebras
• Poisson-Nijenhuis structures on quiver path algebras
• Hochschild cohomology of polynomial representations of $G{L}_2({\bar{\mathbb{F}}}_p)$
• Noncommutative Hamiltonian dynamics on foliated manifolds
• Shifted double Lie-Rinehart algebras
• Dense quasi-free subalgebras of the Toeplitz algebra
• Noncommutative Geometry and dynamical models on U(u(2)) background
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations
• Universal deformation formulas
• A new family of algebras whose representation schemes are smooth