In this lecture, we are going to present two Gelfand-Naimark theorems: Firstly, it is shown that every commutative C*-algebra is isomorphic to the algebra of continuous functions on its spectrum, vanishing at infinity. Finally, a fundamental theorem, known as GNS construction will be discussed. Actually it is shown that every C*-algebra can be seen as a C*-subalgebra of the algebra of bounded linear operators on a Hilbert space.

Gromov width of a symplectic manifold M is a supremum of capacities of balls that can be symplectically embedded into M. The definition was motivated by the Gromov's Non-Squeezing Theorem which says that maps preserving symplectic structure form a proper subset of volume preserving maps.

Let G be a compact connected Lie group G, T its maximal torus, and $\lambda$ be a point in the chosen positive Weyl chamber.

The group G acts on the dual of its Lie algebra by coadjoint action. The coadjoint orbit, M, through $\lambda$ is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width.

In many known cases the width is exactly the minimum over the set of positive results of pairing $\lambda$ with coroots of G:

$$\min \{ \langle \alpha_j^{\vee},\lambda \rangle; \alpha_j \textrm{ a coroot, }\langle \alpha_j^{\vee},\lambda \rangle>0\}.$$

For example, this result holds if G is the unitary group and M is a complex Grassmannian or a complete flag manifold satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system to construct symplectic embeddings of balls. In this way we prove that the above formula gives the lower bound for Gromov width of U(n) and SO(n) coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the case of regular U(n) orbits.

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.