Combinatorics & Definability Properties of Entire Functions

The project brings together the mathematical disciplines of set theory and complex analysis. Georg Cantor developed set theory in the 19th century. One of his most important discoveries was that not all infinite sets have the same size. The smallest infinity is that of the natural numbers, sets of this size are called countable, all other infinite sets are called uncountable. This observation led to the formulation of the statement, known as the continuum hypothesis, that every uncountable set of real numbers is as large as the set of all real numbers. In the 20th century Kurt Gödel and Paul Cohen showed that the continuum hypothesis can neither be proven nor refuted with the normal axioms of set theory. In the last few decades, connections to other areas of mathematics have been established, especially in the theory of sets of real numbers. The continuum hypothesis cannot be proved with the normal axioms of set theory, but this is sometimes possible for weaker statements. Various properties of sets of real numbers have been studied, and it can often be shown that the existence of a set of real numbers with a property X implies the existence of an equally large or smaller set of real numbers with a property Y. The history of complex analysis also goes back to the 19th century and was developed by several mathematicians, including Cauchy, Riemann and Weierstrass. Their objects, the holomorphic functions, can be defined in several equivalent ways, as differentiable functions over the complex numbers, as functions given by power series or as conformal mappings between two-dimensional spaces. There are just as many holomorphic functions as there are real numbers. Paul Erdos showed in 1964 that the continuum hypothesis is equivalent to the statement that there is an uncountable set M of holomorphic functions such that for every complex number z, the set of values that functions from M take on at z is countable. In recent years, the set theory of holomorphic functions has received increased attention, e.g. through the work of Burke, Kumar and Shelah. As part of the project, the set theory of holomorphic functions is to be further developed and expanded to include the perspective of descriptive set theory. Descriptive set theory examines under which conditions objects, such as a certain family of holomorphic functions, whose existence can be proven abstractly using the axioms of set theory, can also be defined.