The lecture will be devoted to the new vintage in the theory of special functions, a unification of the Bessel, hypergeometric, spherical and Whittaker functions, their p-adic and difference counterparts, and of course the theta-functions (associated with root systems) in one definition. The latter was suggested 13 years ago, but a reasonably complete analytic theory of such "global spherical functions" was created only recently, including the Harish-Chandra asymptotic formula and many more. These global functions generalize the classical q-hypergeometric (basic) series introduce by Heine in 1846, but the new approach is very different even for one variable. Algebraically, the global functions are actually similar to the Bessel functions. For instance, the corresponding Fourier transform is essentially involutive (like for the classical Hankel transform), which is broken in the Harish-Chandra theory and its p-adic counterpart. The construction is based on DAHA, q-deformations of the classical Heisenberg-Weyl algebras. The Lie theory was expected to provide such unification (the Gelfand program) but it did not materialize. The new theory actually begins with "nonsymmetric" global spherical functions (eigenfunctions of first order opertators) and their counterparts everywhere (including "nonsymmetric" Schur functions), which is generally beyond the Lie theory. It clarifies why we need to go "back" to the Heisenberg algebras and their deformations. The case of rank one will be mainly considered; no special knowledge of representation theory is assumed.

Quasiconformal mappings $u:\Omega\to \Omega'$  between open domains in $\R^n$,  are $ W^{1,n}$ homeomorphisms whose dilation $K=|du|/ (det du)^1/n$ is in $L^\infty$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within  a given homotopy class or with prescribed boundary data.  In a joint work with A. Raich we study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra. Moment-angle complexes corresponding to simplicial subdivision of spheres are topological manifolds, and those corresponding to simplicial polytopes admit smooth realisations as intersection of real quadrics in C^m.

After an introductory part describing the general properties of moment-angle complexes we shall concentrate on the complex-analytic and Lagrangian aspects of the theory. We show that the moment-angle manifolds corresponding to complete simplicial fans admit nonKaehler complex-analytic structures. This generalises the known construction of complex-analytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds. We proceed by describing the Dolbeault cohomology and certain Hodge numbers of moment-angle manifolds by applying the Borel spectral sequence to holomorphic principal bundles over toric varieties.

A new wide family of minimal Lagrangian submanifolds N in C^m or CP^m can be constructed from intersections of real quadrics. These submanifolds have the following topological properties: every N embeds in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a small cover with fibre a torus. These properties are used to produce new examples of Lagrangian submanifolds with quite complicated topology. Different parts of this talk are based on joint works with Victor Buchstaber, Andrei Mironov and Yuri Ustinovsky.