Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

This monograph presents various ongoing approaches to the vast topic of quantization, which is the process of forming a quantum mechanical system starting from a classical one, and discusses their numerous fruitful interactions with mathematics.

Quantization, Geometry and Noncommutative Structures in Mathematics and Physics (A. Cardona, H. Ocampo, P. Morales, S. Paycha, A.F. Reyes Lega (Eds.)).- General Overview (Alexander Cardona, Sylvie Paycha and Andrés F. Reyes Lega).- Introduction.- Poisson Geometry and Classical Dynamics.- Geometric and Deformation Quantization.- Noncommutative Geometry and Quantum Groups.- Deformation Quantization and Group Actions (Simone Gutt).- What do we mean by quantization?.- Deformation Quantization.- Fedosov's star products on a symplectic manifold.- Classification of Poisson deformations and star products.- Star products on Poisson manifolds and formality.- Group actions in deformation quantization.- Reduction in deformation quantization.- Some remarks about convergence.- . Principal fiber bundles in non-commutative geometry (Christian Kassel).- Introduction.- Review of principal fiber bundles.- Basic ideas of non-commutative geometry.- From groups to Hopf algebras.- Quantum groups associated with SL2(C).- Group actions in non-commutative geometry.- Hopf Galois extensions.- Flat deformations of Hopf algebras.- An Introduction to Nichols Algebras (Nicolás Andruskiewitsch).- Preliminaries.- Braided tensor categories.- Nichols algebras.- Classes of Nichols algebras.- Quantum Field Theory in Curved Space-Time (Andrés F. Reyes Lega).- Introduction.- Quantum Field Theory in Minkowski Space-Time.- Quantum Field Theory in Curved Space-Time.- Cosmology.- An Introduction to Pure Spinor Superstring Theory (Nathan Berkovits and Humberto Gomez).- Introduction.- Particle and Superparticle.- Pure Spinor Superstring.- Appendix.- Introduction to Elliptic Fibrations (Mboyo Esole).- Introduction.- Elliptic curves over C.- Elliptic fibrations.- Kodaira-Néron classification of singular fibers.- Miranda models.- Batalin-Vilkovisky formalism as a theory of integration for polyvectors (Pierre J. Clavier and Viet Dang Nguyen).- Motivations and program.- BV integral.- Gauge fixing.- Master equations.- Conclusion.- Split Chern-Simons theory in the BV-BFV formalism (Alberto S. Cattaneo, Pavel Mnev, and Konstantin Wernli).- Introduction.- Overview of the BV and BV-BFV formalisms.- Chern-Simons theory as a BF-like theory.- Split Chern-Simons theory on the solid torus.- Conclusions and outlook.- Weighted direct product of spectral triples (Kevin Falk).- Introduction and motivation. -Weighted direct product of spectral triples.- Example of weighted direct product with Toeplitz operators.- Index.