In this lecture we are going to talk about an equivalence between the category of compact Hausdorff topological spaces and the category of commutative unital C*-algebras. We also introduce spectrum of an element in a unital C*-algebra and give some examples. Then we present GNS construction.

For a symmetric monoidal model category $\mathscr{M} = (\underline{M};\otimes; I)$, we develop a theory of weakly enriched categories over $\mathscr{M}$, called co-Segal $\mathscr{M}$-categories. By ‘weakly enriched’ we mean a sort of enriched category where there is no prescription of a composition; but rather we allow many possible compositions, where each of them is associative up-to homotopies. Their definition derives from the philosophy of classical Segal categories; and just like for Segal categories they give rise to higher categorical structures. In this talk I will present the theory of co-Segal categories and will give the first results on their homotopy theory.

In this series of 3 talks I will first state the general concept of a system of PDE's associated to a non-degenerate CR-manifold. The idea goes back to Lie, Cartan and Segre, and it was undeservedly forgotten. The PDE-approach was recently reviewed by A.Sukhov and J.Merker and enabled the latter one to obtain some interesting results in CR-geometry. The classical results of S.Lie and the recent results of J.Merker will be stated on the second lecture. Finally, on the last lecture I will tell about a recent result with R.Shafikov concerning extension of holomorphic mappings where the PDE-approach was successfully applied as well.

One of the main themes of invariant theory is to relate the $G$-invariant regular functions, of a regular action $G\times X\to X$, to some suitable quotient morphism $\pi : X\to Y$. However, there are examples to show that the naive attempt $X\mapsto k[X]^G$ does not lead directly to any appealing conclusion. Indeed, $k[X]^G$ may not be finitely generated, or it may not be "large" enough to separate the $G$-orbits of $G\times X\to X$, even generically.

The purpose of this talk is to discuss some basic results of "quasi-invariant theory". The main ideas here have their roots in the work of Hilbert, Zariski, Nagata, and Rosenlicht. Our major purpose is to assess the influence of quasi-invariant rational functions and $G$-invariant divisors on the problem of constructing a useful quotient object of a regular action $G\times X\to X$.