Concern about Poincaré Normal Form of Riemann matrices

[Jami]

Quick Intro: Myself Mahraib Fatima, a third-year Computer Science undergraduate and self-paced Machine Learning student, with interest about algebraic geometry and complex analysis.

Previous Knowledge: I have studied complex analysis, algebraic geometry, and Jacobian varieties, which align with the mathematical foundations required for this project.

Project Research: Since February, I explored SageMath's project list and focused on "Poincaré Normal Form of Riemann matrices." I thoroughly read and analyzed the research paper on Poincaré's Complete Reducibility Theorem, abelian integrals, Riemann surfaces, and topological graphs.

Learning: I learnt  about Riemann matrices, theta functions, and their reducibility, as described in the paper.

Implementation Goal: The project aims to implement Poincaré's theorem in SageMath, focusing on decomposing Riemann matrices and reducing associated theta functions(As mentioned in project description).

Ambiguity: The main challenge lies in translating the paper's theoretical algorithms into efficient code, especially handling edge cases and ensuring numerical stability (mentor guidance required).

Approach: I plan to break the implementation into modular steps: Riemann matrix decomposition, theta function reduction, and testing.

Tools: I will use Python, SageMath, and libraries like NumPy and SymPy for computations and symbolic mathematics.

Outcome: The project will provide a computational tool for researchers and students, along with documentation and examples.

Motivation: This project combines my interests in mathematics and programming, allowing me to contribute to the SageMath community while deepening my knowledge.

Regards,

Mahraib Fatima