Dear SageMath Team,
I hope this email finds you well. My name is *Ahmad Faraz*, and I am a
*second-year
undergraduate student, studying Computer Science and Engineering at Amity
University Jharkhand, India,* with a strong interest in differential
geometry and open-source software development. I came across the "Implement
a Solver for the Killing Equations" project idea listed under SageMath for
GSoC 2025, and I am excited about the opportunity to contribute to this
mathematically rich and challenging endeavor.
I have experience with Python, symbolic computation (SymPy), and a solid
background in PDEs and tensor analysis, which I believe align well with the
project’s requirements. I am particularly motivated by the prospect of
enhancing SageMath’s capabilities in differential geometry, a field I find
both intellectually stimulating and practically impactful. I would love to
discuss this project further with you, share my draft proposal, and seek
your feedback to refine my approach before the application deadline.
*GSoC Application Proposal Personal:*
*Name*: ahmad Faraz
*Contact Information*:
- Email: [ahmadfaraz00...@gmail.com]
- *Location/Timezone*: Jharkhand, India / UTC-05:30
*University*: Second-year undergraduate in Computer Science and
Engineering at Amity University, Jharkhand with 9.73 CGPA.
*Background:* I am a second-year undergraduate with a strong foundation in
mathematics and programming. My technical skills include:
- *Programming*: Proficient in Python, experience with symbolic
computation (SymPy), and familiarity with SageMath.
- *Mathematics*: Comfortable with advanced calculus, linear algebra,
PDEs, and differential geometry (e.g., tensor analysis, Riemannian
manifolds). I’ve completed coursework in general relativity, where I
studied Killing vectors.
- *Experience*: Developed a Python-based tool for solving PDEs as a
personal project; contributed minor bug fixes to open-source project
I’m passionate about mathematical software because it bridges abstract
theory and practical computation, making complex ideas accessible. This
project excites me due to its blend of geometry, PDEs, and coding—areas
where I thrive and want to grow.
- *Open-Source Engagement*: Limited but growing—small contributions to
open-source projects.
- *Pet Projects*: Built a PDE solver for heat equations; currently
exploring tensor computations in Python.
- *SageMath Experience*: User for 6 months, initially for linear
algebra, now exploring its differential geometry module.
------------------------------
Project:
*Title*: Implementing a Solver for the Killing Equations in SageMath
*Length*: Long (350 hours)
*Project Synopsis*:
This project aims to develop a solver within SageMath for the Killing
equations, a system of PDEs that determine Killing vector fields on
pseudo-Riemannian manifolds. These fields, which preserve the metric under
their flow, are crucial in geometry and physics (e.g., symmetries in
relativity). The solver will take a user-defined metric and coordinates,
compute the necessary geometric objects (e.g., Christoffel symbols), and
return the Killing vectors, enhancing SageMath’s differential geometry
toolkit.
*Personal Involvement*:
I’ve been fascinated by Killing vectors since studying general relativity,
where they reveal spacetime symmetries. Contributing to SageMath aligns
with my goal to deepen my expertise in geometry while advancing open-source
tools for the mathematical community.
*Details*:
The project breaks into modular tasks with deliverables:
1. *Metric Input and Setup (40 hours)*
- Task: Extend SageMath’s manifold module to accept a user-defined
metric and coordinates.
- Deliverable: A function define_manifold(coords, metric) returning a
manifold object.
- Result: Users can input, e.g., flat 2D metric or Minkowski
spacetime.
2. *Christoffel Symbol Computation (50 hours)*
- Task: Implement or optimize a routine to compute Γμνλ from the
metric.
- Deliverable: Method christoffel_symbols() integrated into the
manifold object.
- Result: Accurate connection coefficients for any input metric.
3. *Killing Equations Formulation (60 hours)*
- Task: Construct the PDE system ∇μKν+∇νKμ=0.
- Deliverable: Function killing_equations() generating symbolic
equations.
- Result: Equations ready for solving, e.g., 3 PDEs in 2D, 10 in 4D.
4. *Symbolic Solver Integration (80 hours)*
- Task: Use SageMath’s symbolic engine (or extend it) to solve the
PDEs for Kμ.
- Deliverable: Method solve_killing() returning Killing vector
components.
- Result: Solutions like K=(a,b) for flat space, validated
analytically.
5. *Testing and Validation (70 hours)*
- Task: Test on manifolds (Euclidean, spherical, Minkowski); verify LK
g=0.
- Deliverable: Test suite with at least 5 manifolds and documentation.
- Result: Reliable, documented solver.
6. *Optimization and User Interface (50 hours)*
- Task: Optimize performance; add a user-friendly interface (e.g.,
Jupyter examples).
- Deliverable: Polished module with tutorial notebook.
- Result: Accessible tool for SageMath users.
*Schedule*:
- *March 31 - April 15, 2025 (Community Bonding)*: Familiarize with
SageMath codebase, discuss with mentor.
- *April 16 - May 15 (40h)*: Task 1 – Metric setup.
- *May 16 - June 15 (60h)*: Task 2 – Christoffel symbols (exams May
20-25, reduced hours).
- *June 16 - July 15 (80h)*: Task 3 – Equations; mid-term prep.
- *July 16 - August 15 (100h)*: Task 4 – Solver; Task 5 – Testing.
- *August 16 - September 1 (70h)*: Task 6 – Optimization, docs; final
submission.
*Risk Management*:
- *Risk*: PDE solver fails for complex metrics.
- Mitigation: Start with simple cases (flat space); fallback to
numerical hints if symbolic fails.
- *Risk*: Time overrun on symbolic solving.
- Alternative: Deliver partial solver (e.g., 2D only) and document
extension steps.
- *Risk*: Bugs in Christoffel computation.
- Mitigation: Cross-check with known results; add unit tests early.
I’m based in UTC-05:30, and I’d be grateful for any insights you could
provide on the project’s scope or implementation details within SageMath.
Thank you for your time and consideration. I look forward to the
possibility of collaborating with the SageMath community!
Best regards,
*Ahmad Faraz *ahmadfaraz00...@gmail.com https://github.com/Shevilll
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