Abstract: I shall give a quick survey of some major recent results in noncommutative geometry. They are mostly related to analytic/geometric aspects of spectral triples and their applications.

Abstract: Given a monoidal model category $M$, we introduce a theory of co-Segal $M$-categories which are homotopy enriched categories over $M$. Examples of such categories emerge naturally when we consider homotopy transfers of algebraic structure. In this talk I will present the theory along with some examples and then will focus on the homotopy theory of these structures. Different model structures for co-Segal categories exist and I will talk about the canonical model structure, which is somehow the correct one.