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.

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.

First we shall present some basic facts about condenser capacity, Green functions and their relation with complex analysis. Then we will examine the asymptotic behavior of the capacity of the inverse image of a condenser under exponential Blaschke products and universal covering maps.

After introducing classical Chern-Simons gauge theory, I will go over the noncommutative version of the theory and also the recent results on computing the action explicitly on some non-commutative spaces.

Gromov's theorem on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of $\mathbb C^n$ has had a deep impact on symplectic topology and complex analysis. I will discuss generalizations of this result to singular submanifolds.

I will explain how permutation modules can be used to compute the motive of a geometrically rational surface. As a by-product one gets that a geometrically split motive with rational coefficients is always 0-dimensional.