Basic definitions and terminology, statement of the main results: Embedding Theorem, Theorem on realization of ${\mathrm{KK}(\mathcal{C}; \cdot,\cdot)}$) by C*-morphisms, On applications.

A generalized Weyl-von-Neumann Theorem in the spirit of Voiculescu and Kasparov, Actions of topological spaces on C*-algebras versus matrix operator convex cones $\mathcal{C}$, related "universal" Hilbert bi-modules, Cone ${\mathcal{C}}$-dependent Ext-groups $\mathrm{Ext}(\mathcal{C};\, A,B)$, Related semi-groups.

Ideal system equivariant Embedding Theorem (II) (Lecture Notes)

C*-systems and its use for embedding results, the example of embeddings into $\mathcal{O}_2$, criteria for existence of ideal equivariant liftings, i.e. characterization of invertible elements in the extension semigroup.

Proof of a special case by construction of a suitable C*-system, Outline of the idea for the proof of the general case by study of asymptotic embeddings, using continuous versions of Rørdam semi-groups.

Operations on the class of s.p.i. algebras, coronas and asymptotic algebras of strongly purely infinite algebras, tensorial absorption of $\mathcal{O}_\infty$, 1-step innerness of residually nuclear c.p. maps.

Definition and properties of the natural group epimorphism from the $\mathcal{C}$-dependent Rørdam group R($\mathcal{C};\, A,B)$ onto Ext($\mathcal{SC}; A; SB$), reduction of the isomorphism problem to the question on homotopy invariance of R($\mathcal{C};\, A,B)$, Some cases of automatic homotopy invariance: the "absorbing" zero element.

Homotopy invariance of R($\mathcal{C};\, A,B)$, existence of C*-morphisms $\varphi:A \rightarrow B$ that represent the elements of R($\mathcal{C};\, A,B)$, proof of the Embedding Theorem in full generality.

Definition and basic properties of $\mathcal{C}$-related ($\mathbb{Z_2}$-graded) Kasparov groups KK($\mathcal{C};\, A,B$) for graded m.o.c. cones $\mathcal{C}$, the isomorphisms Ext($\mathcal{C};\, A,B$) $\cong$ KK($\mathcal{C_{(1)}};\, A,B_{(1)}$) and Ext($\mathcal{SC};\, A,SB$) $\cong$ KK($\mathcal{C};\, A,B$) in trivially graded case. Homotopy invariance of Ext($\mathcal{SC};\, A,SB$). The isomorphism Ext($\mathcal{SC};\, A,SB$) $\cong$ R($\mathcal{C};\, A,B$).

The $KK_{X}(A;B)$ := KK($\mathcal{C_{X}};\, A,B$) classification for X $\cong$ Prim(A) $\cong$ Prim(B), where A, B are stable amenable separable C*-algebras.Structure of the algebras with ideal-system preserving zero-homotopy.

Some conclusions of the classication results and open questions (Lecture Notes)

Constructions of examples of algebras with given second countable locally compact sober $T_0$ spaces (not necessarily Hausdorff). Minimal requirement for a weak version of a universal coefficent theorem for ideal-equivariant classication, indications of possible equivariant versions for actions of compact groups (up to 2-cocycle equivalence).