Workshops on the frontiers of Nevanlinna theory

Complex analysis is the extension of calculus to the complex numbers. Meromorphic functions are functions that are differentiable throughout the complex plane except possibly at isolated points where the functions may have the simplest kind of singularities, called poles. Such functions arise naturally in many theoretical and practical problems. Britain has a strong tradition of research in function theory. Among its particular strengths are Nevanlinna's value distribution theory for meromorphic functions of a single variable, as well as the study of dynamical systems. However, during the last couple of decades there have been major breakthroughs in Nevanlinna theory and related areas which appear to have had little impact in Britain to date. Funds are sought to run a series of small workshops that are aimed at engaging the UK community with some of these important lines of research. The emphasis of the workshops will be on exploring collaborations and the number of talks will be restricted accordingly.Below are brief descriptions of each proposed workshop.W1. Frontiers of Nevanlinna theoryThis will be a general introduction to the main themes.W2. Nevanlinna theory and Diophantine approximationThis workshop will explore the remarkable formal connection between Nevanlinna theory and an area of number theory. Participants will be invited who work on this connection as well as those working in pure Nevanlinna theory or Diophantine approximation.W3. Function theory and dynamical systems over p-adic spacesFor each prime number p, the p-adic numbers are a field of numbers that contain the rational numbers (fractions) but are quite different in nature to the real numbers. They arise naturally in many problems in number theory. One can also construct analogues of the complex numbers in the p-adic setting and develop large parts of complex analysis, including Nevanlinna theory. Recently there has been a lot of interest in the behaviour of dynamical systems in the p-adic setting. Classical Nevanlinna theory has been a useful tool in dynamical systems over the (genuine) complex numbers. This workshop will explore this connection over the p-adics.W4. Function theory of differential and difference equationsThere are several important conjectures concerning differential and difference equations in the complex domain for which Nevanlinna theory would be a useful tool.

Several important problems from otherwise apparently unrelated areas lead to questions that can be addressed by Nevanlinna theory. One such problem, which connects with many areas of applied mathematics, physics and probability, as well as pure mathematics, concerns a conjecture of Boris Dubrovin. The conjecture concerns the existence of a solution of a certain differential equations with a particular sector free of poles. This conjecture implies the validity of an improved asymptotic representation of solutions of some important wave equations. This phenomenon is expected to be universal in some sense. This is a problem for which sectorial Nevanlinna theory is a promising tool. As well as other problems in dynamical systems, there are problems in Diophantine approximation that are to be addressed by analogy . Vojta's dictionary provides a formal connection between the main definitions and theorems of Nevanlinna theory and Diophantine approximation. This is a heuristic, not a rigorous correspondence, but it has led to progress primarily by rewriting and then extending part of one theory to look more like the other. It has led to the development of new tools in addressing new questions. For example, Serge Lang and others since him, began to explore the error terms that arise in Diopahntine approximation in such a way that mimics the corresponding estimates in Nevanlinna theory. A strong analogy between the error and ramification terms in Nevanlinna theory with corresponding terms in Diophantine approximation would give the famous ABC conjecture, from which the theorem of Wiles ( Fermat's Last theorem ) follows trivially.