Project VD

Formalizing Value Distribution Theory

This project aims to formalize value distribution theory for meromorphic functions in the complex plane, roughly following the sections on Nevanlinna Theory in Serge Lang's "Introduction to Complex Hyperbolic Spaces". The project uses the lean4 computer language and builds on the lean mathematical library.

We are looking for collaborators. Please be in touch if you would like to join the fun!

State of Affairs and Future Plans

With the formalization of "Nevanlinna's First Main Theorem", the project has recently reached its first milestone. The current code has "proof of concept" quality: It compiles fine but needs refactoring and documentation. The next goals are as follows.

• Clean up the existing codebase and feed intermediate results into the lean mathematical library
• Formalize the Theorem on Logarithmic Differentials
• Formalize the Second Main Theorem of value distribution theory
• Establish some of the more immediate applications

These plans might change, depending on feedback from the community and specific interests of project participants.

Material Covered

We touched on the following subjects.

• Harmonic functions in the complex plane
  • Laplace operator and associated API
  • Definition and elementary properties of harmonic functions
  • Mean value properties of harmonic functions
  • Real and imaginary parts of holomorphic functions as examples of harmonic functions
• Holomorphic functions in the complex plane
  • Existence of primitives [duplication of work already under review at mathlib]
  • Existence of holomorphic functions with given real part
• Meromorphic Functions in the complex plane
  • API for continuous extension of meromorphic functions, normal form of meromorphic functions up to changes along a discrete set
  • Behavior of pole/zero orders under standard operations
  • Zero/pole divisors attached to meromorphic functions and associated API
  • Extraction of zeros and poles
• Integrals and integrability of special functions
  • Interval integrals and integrability of the logarithm and its combinations with trigonometric functions; circle integrability of log ‖z-a‖
  • Circle integrability of log ‖meromorphic‖
• Basic functions of Value Distribution Theory
  • The positive part of the logarithm, API, standard inequalities and estimates
  • Logarithmic counting functions of divisors
  • Nevanlinna heights of entire meromorphic functions
  • Proximity functions for entire meromorphic functions
• Jensen's formula
• The First Main Theorem