The Noncommutative Geometry of Real Calculi

Abstract: Noncommutative geometry extends the traditional connections between algebra and geometry beyond the realm of commutative algebras, allowing for a broader exploration of geometric concepts in noncommutative settings. The geometric perspective facilitates the study and understanding of various mathematical structures, including operator algebras, quantum groups, and noncommutative spaces. Since its inception, noncommutative geometry has experienced remarkable growth, attracting mathematicians from diverse backgrounds who seek to delve into the geometric aspects of noncommutative structures. Through this lens, groundbreaking discoveries have deepened our understanding of fundamental mathematical principles and opened up new avenues of research. This ongoing exploration not only enriches our mathematical knowledge but also finds practical applications in theoretical physics, quantum field theory, and interdisciplinary fields.The primary focus of this thesis is to offer valuable insights into the derivation-based approach of real calculi, which employs modules over an algebra as an algebraic analogy for vector bundles over differential manifolds. An overarching goal is to give noncommutative counterparts of classical geometric concepts, with a specific emphasis being placed on a noncommutative adaptation of the Levi-Civita connection in (pseudo-)Riemannian geometry. An investigation into the existence of a Levi-Civita connection is conducted in the context of general projective modules, and in cases where it exists a theory of embeddings is developed and used to give a minimal embedding of the noncommutative torus into the noncommutative 3-sphere. The thesis also establishes the concept of morphisms of real calculi, which plays a crucial role in examining the relationship between projective modules and specific free modules in this framework. Moreover, the thesis provides an in-depth examination of matrix algebras, utilizing them as illustrative examples to showcase the process of determining isomorphism classes of real calculi in various scenarios and presenting classes of examples where a Levi-Civita connection does not exist.