Harmonic and Complex Analysis and its Applications

This article is meant as both an introduction and a review of some of the recent developments on Fock and Bergman spaces of polyanalytic functions. The study of polyanalytic functions is a classic topic in complex analysis. However, thanks to the interdisciplinary transference of knowledge promoted within the activities of HCAA network it has benefited from a cross-fertilization with ideas from signal analysis, quantum physics, and random matrices. We provide a brief introduction to those ideas and describe some of the results of the mentioned cross-fertilization. The departure point of our investigations is a thought experiment related to a classical problem of multiplexing of signals, in order words, how to send several signals simultaneously using a single channel.

In this paper we present a historical and scientific account of the development of the theory of the Löwner–Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Löwner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory.

The Schwarz Lemma has given impetus to developments in several areas of complex analysis and mathematics in general. We survey some investigations related to its three parts (invariance, rigidity, and distortion) that began early in the twentieth century and are still being carried out. We consider only functions analytic in the unit disk. Special attention is devoted to the Boundary Schwarz Lemma and to applications of the Schwarz–Pick Lemma and the Boundary Schwarz Lemma to modern rigidity theory and complex dynamics.

Coorbit theory arose as an attempt to describe in a unified fashion the properties of the continuous wavelet transform and the STFT (Short-time Fourier transform) by taking a group theoretical viewpoint. As a consequence H.G. Feichtinger and K.H. Gröchenig have established a rather general approach to atomic decomposition for families of Banach spaces (of functions, distributions, analytic functions, etc.) through integrable group representations (see Feichtinger and Gröchenig (Lect. Notes Math. 1302:52–73, 1988; J. Funct. Anal. 86(2):307–340, 1989; Monatsh. Math. 108(2–3):129–148, 1989), Gröchenig (Monatsh. Math. 112(3):1–41, 1991)), now known as coorbit theory.They gave also examples for the abstract theory and until now this approach gives new insights on atomic decompositions, even for cases where concrete examples can be obtained by other methods. Due to the flexibility of this theory the class of possible atoms is much larger than it was supposed to be in concrete cases. It is a remarkable fact that almost all classical function spaces in real and complex variable theory occur naturally as coorbit spaces related to certain integrable representations.In the present paper we present an overview of the general theory and applications for the case of the weighted Bergman spaces over the unit disc, indicating the benefits of the group theoretic perspective (more flexibility, at least at a qualitative level, more general atoms).

This survey describes recent advances on quadrature domains that were made in the context of the ESF Network on Harmonic and Complex Analysis and its Applications (2007–2012). These results concern quadrature domains, and their two-phase counterparts, for harmonic, subharmonic and analytic functions.

We give an overview of some recent developments concerning harmonic and other moments of plane domains, their relationship to the Cauchy and exponential transforms, and to the meromorphic resultant and elimination function. The paper also connects to certain topics in mathematical physics, for example domain deformations generated by harmonic gradients (Laplacian growth) and related integrable structures.

Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general multi-point situation with arbitrary splittings. Here we will review the path of developments from the Virasoro algebra to its higher genus and multi-point analogs. The starting point will be a Poisson algebra structure on the space of meromorphic forms of all weights. As sub-structures the vector field algebras, function algebras, Lie superalgebras and the related current algebras show up. All these algebras will be almost-graded. In detail almost-graded central extensions are classified. In particular, for the vector field algebra it is essentially unique. The defining cocycle is given in geometric terms. Some applications, including the semi-infinite wedge form representations are recalled. We close by giving some remarks on the Lax operator algebras introduced recently by Krichever and Sheinman.