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### Formalizing Value Distribution Theory ### 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. 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! We are looking for collaborators. Please be in touch if you would like to join
the fun!
## State of Affairs and Future Plans ## 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 lean code has "proof of concept" quality: It compiles fine but needs refactoring and documentation. The next goals are as follows. 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 many of the intermediate results into the lean mathematical library - Clean up the existing codebase and feed intermediate results into the lean
mathematical library
- Formalize the Theorem on Logarithmic Differentials - Formalize the Theorem on Logarithmic Differentials
- Formalize the Second Main Theorem of Nevanlinna Theory - Formalize the Second Main Theorem of value distribution theory
- Establish some of the more immediate applications - Establish some of the more immediate applications
These plans might change, depending on feedback from the community and specific interests of project participants. These plans might change, depending on feedback from the community and specific
interests of project participants.
## Material Covered ## Material Covered
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- Laplace operator and associated API - Laplace operator and associated API
- Definition and elementary properties of harmonic functions - Definition and elementary properties of harmonic functions
- Mean value properties of harmonic functions - Mean value properties of harmonic functions
- Real and imaginary parts of holomorphic functions as examples of harmonic functions - Real and imaginary parts of holomorphic functions as examples of harmonic
functions
- Holomorphic functions in the complex plane - Holomorphic functions in the complex plane
- Existence of primitives - Existence of primitives [duplication of work already under review at
mathlib]
- Existence of holomorphic functions with given real part - Existence of holomorphic functions with given real part
- Meromorphic Functions in the complex plane - Meromorphic Functions in the complex plane
- API for continuous extension of meromorphic functions and resolution of resolvable singularities, normal form of meromorphic functions up to changes along a discrete set - 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 - Behavior of pole/zero orders under standard operations
- Zero/pole divisors attached to meromorphic functions and associated API - Zero/pole divisors attached to meromorphic functions and associated API
- Extraction of zeros and poles - Extraction of zeros and poles
- Integrals and integrability of special functions - 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‖ - Interval integrals and integrability of the logarithm and its combinations
with trigonometric functions; circle integrability of log ‖z-a‖
- Circle integrability of log ‖meromorphic‖ - Circle integrability of log ‖meromorphic‖
- Basic functions of Value Distribution Theory - Basic functions of Value Distribution Theory
- The positive part of the logarithm, API, standard inequalities and estimates - The positive part of the logarithm, API, standard inequalities and
- Definition and main properties of the (logarithmic) counting function of a divisor estimates
- Definition and main properties of the characteristic function (=Nevanlinna height function) for an entire meromorphic function - Logarithmic counting functions of divisors
- Definition and main properties of the proximity function (=Nevanlinna height function) for entire meromorphic functions - Nevanlinna heights of entire meromorphic functions
- Proximity functions for entire meromorphic functions
- Jensen's formula - Jensen's formula
- The First Main Theorem - The First Main Theorem