The project

Modern complex analysis lies at the crossroad of many different theories such as function spaces, operator theory, Fourier analysis, subelliptic PDEs, complex and CR geometry and more. The interplay of different techniques from these areas makes complex analysis an extremely rich field in which many new ideas and approaches appear and become fruitful not just for complex analysis, but also for all mathematics and applications.

This project will be developed through very intense collaborations with the young researchers and a network of international collaborations. All participants to the project will have a strong commitment to the project. We aspect to train several PhD students and 8-10 postdocs. We also intend to disseminate our knowledge to a wide-open scientific community. We will contribute to the advancement and consolidation of the Complex Analysis research community at the European level. By the end of the project, we expect to have a fruitful working relation between our team and a number of researchers in engineering, machine learning, physics, and life sciences.

At the center of the project there will be a website and a protocol for online communication. The website will have a public part and one reserved to the project members. The public window will contain a repository of preprints, seminars, lecture series, announcements of project related seminars and workshops. We will work with other groups of researchers in Italy and worldwide and we will participate in the organization of conferences and workshops, and other initiatives. Part of the public window will be devoted to the popularization of complex analysis, its problems, history, and applications, to a wide public of researchers, students, stakeholders (public and private organizations and enterprises), and citizens. Part of the content will be in English, part in Italian, depending on the public it is aimed at. Part of the website will be accessible to the members of the project only and it will contain material related to ongoing projects, including notes and videos of internal meetings.

There will be a monthly online meeting of all members of the project, which will sometimes have the form of a research workshop.

The research groups within the projects will mostly meet online. We plan to have a project's workshop "live" each year allowing remote attendance. The activities of the PRIN group will be interlaced in several ways with those of the international research community working in Complex Analysis. In particular, we will advertise the postdoctoral positions in a large community, with aim of having in Italy some of the best recent PhD's in the area from Europe, and beyond. We will invite experts from abroad, to jointly do research and to deliver seminars and series of lectures, sometimes in the form of a PhD course, which will be hosted by one university in the project, but open to online participation from other universities. We will organize mini-workshops devoted to a well-defined topic, or problem, with the goal of favoring collaboration and exchange.

Main topics of interest:

(1) Holomorphic function spaces and reproducing kernel Hilbert spaces

(1a) Paley-Wiener and de Branges spaces in one and several variables

(1b) Dirichlet spaces and multi-parameter potential theory

(1c) Drury-Arveson space and multivariable operator theory

(1d) Weighted Bergman spaces on homogeneous Siegel domains

(2) Analytic and geometric properties of domains in Cn

(2a) Analysis on worm domains

(2b) Domains admitting complex submanifolds in their boundary

(2c) Analysis on singular domains

(3) CR geometry and PDEs

(3a) CR immersions

(3b) Subelliptic harmonic maps

(3c) Testing families of analytic disks

(3d) L^p-functional calculus of subLaplacians

(3e) Riemann-Hilbert correspondence and Fourier transforms

(4) Holomorphic mappings and complex dynamics

(4a) One-parameter semigroups of holomorphic self-maps

(4b) Loewner theory

(5) Hypercomplex function theory

(5a) Function theory of slice-regular functions

(5b) Quaternionic manifolds

(5c) Monogenic functions and CR geometry


Applications of complex analysis to other mathematical disciplines, natural sciences, and technology, have an uninterrupted history dating from the inception of the theory. This concerns the results, as well as the methods. At present in Italy there is little communications between researchers developing complex analysis and scientists and technicians using it. One of the main purposes of our research projects is bridging this gap.

We plan to work on a series of applications, collaborating with researchers from other disciplines. We plan to make the collaboration stronger and more effective by recruiting some PhD students and/or postdocs with a previous training in physics, or engineering.

Main applications connected to the project. 

Analysis of visual cortex via the Citti-Sarti model

Space-time singularities, quantization of mechanical systems

Stochastic processes

Computer-aided geometric design

Signal theory, Nyquist-rate, control theory

Machine learning