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This dissertation by Axel Tiger Norkvist explores advanced mathematical concepts in the field of geometry, particularly focusing on the geometry of real calculi. It extends traditional algebraic and geometric connections beyond commutative algebras, facilitating the study of complex mathematical structures such as operator algebras and quantum groups. The work delves into the adaptation of the Levi-Civita connection within general projective modules and develops a theory of embeddings, including the minimal embedding of the torus into the noncommutative 3-sphere. Additionally, it introduces the concept of morphisms of real calculi, examining their role in the relationships between projective modules. This research contributes to a deeper understanding of fundamental mathematical principles and opens new avenues for theoretical exploration and practical applications in science and technology.

Noncommutative algebras. --- Geometry, Differential.

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Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, the authors prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and the authors illustrate this point with two examples in the text. In order to understand what is new in their approach in the commutative setting the authors prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds their index formula appears to be completely new.

Geometry. --- Index theory (Mathematics) --- Noncommutative algebras.

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Noncommutative algebras --- Semigroups --- Algèbres non commutatives --- Semi-groupes

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Geometry, Algebraic --- Noncommutative algebras --- Géométrie algébrique --- Algèbres non commutatives --- Noncommutative rings

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Factorization algebras are local-to-global objects that play a role in classical and quantum field theory that is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this second volume, the authors show how factorization algebras arise from interacting field theories, both classical and quantum, and how they encode essential information such as operator product expansions, Noether currents, and anomalies. Along with a systematic reworking of the Batalin-Vilkovisky formalism via derived geometry and factorization algebras, this book offers concrete examples from physics, ranging from angular momentum and Virasoro symmetries to a five-dimensional gauge theory.

Quantum field theory --- Noncommutative algebras. --- Geometric quantization. --- Factors (Algebra) --- Factorization (Mathematics) --- Mathematics.