Geometric Langlands CLI

A user-friendly command-line interface for the Geometric Langlands computational framework, making advanced mathematical computations accessible to researchers and mathematicians.

Features

• Interactive REPL for mathematical computations
• Batch processing of correspondences and verifications
• Rich visualizations of mathematical objects
• Persistent storage with SQLite database
• Multiple export formats (JSON, LaTeX, Mathematica, SageMath, Python)
• Configuration management with TOML files
• Progress tracking with detailed progress bars

Installation

# Install from source
git clone https://github.com/ruvnet/ruv-FANN.git
cd ruv-FANN/geometric_langlands_conjecture/geometric-langlands-cli
cargo install --path .

Quick Start

Start Interactive REPL

langlands repl

Compute Correspondences

# Verify Langlands correspondence
langlands compute correspondence --input "GL(3)" --output results.json

# Compute Hecke eigenvalues
langlands compute hecke --input "GL(2)" --parallel

# Evaluate L-functions
langlands compute l-function --input "eisenstein" --output l_values.csv

Visualizations

# Visualize Hecke eigenvalues
langlands visual hecke-eigenvalues --interactive --resolution 1920x1080

# Plot L-function
langlands visual l-function --output l_function.png

# Visualize moduli space
langlands visual moduli-space --interactive

Train Neural Networks

# Train on correspondence patterns
langlands train --dataset training_data.json --epochs 100 --save-model model.bin

# Use custom architecture
langlands train --dataset data.json --architecture deep --learning-rate 0.001

Verify Mathematical Properties

# Verify Ramanujan conjecture
langlands verify ramanujan --input "GL(2)" --depth deep --proof

# Check functoriality
langlands verify functoriality --input "GL(2)->GL(3)" --proof

# Verify reciprocity laws
langlands verify reciprocity --depth exhaustive

Export Results

# Export to LaTeX
langlands export recent --format latex --output paper.tex --metadata

# Export to Mathematica
langlands export computation_id --format mathematica --output notebook.nb

# Export to Python
langlands export all --format python --output analysis.py

Commands

Core Commands

• langlands compute <type> - Run mathematical computations
• langlands visual <type> - Create visualizations
• langlands train - Train neural networks
• langlands verify <property> - Verify mathematical properties
• langlands export <source> - Export results in various formats
• langlands repl - Start interactive session

Computation Types

• correspondence - Langlands correspondence verification
• hecke - Hecke operator eigenvalues
• l-function - L-function evaluations
• trace-formula - Trace formula computations
• spectral - Spectral decomposition
• functoriality - Functorial lifts
• ramanujan - Ramanujan conjecture verification

Visualization Types

• sheaf - Perverse sheaf structure
• representation - Galois representations
• moduli-space - Moduli space of bundles
• spectral-curve - Spectral curves
• hecke-eigenvalues - Hecke eigenvalue plots
• l-function - L-function plots
• correspondence - Langlands correspondence diagram

Verification Properties

• correspondence - Langlands correspondence
• functoriality - Functorial properties
• reciprocity - Reciprocity laws
• ramanujan - Ramanujan conjecture
• selberg - Selberg trace formula
• riemann-hypothesis - Generalized Riemann hypothesis
• local-global - Local-global principle

Database Management

# Initialize database
langlands db init

# List stored computations
langlands db list --limit 20

# Show computation details
langlands db show <computation_id>

# Export/import database
langlands db export database_backup.json
langlands db import database_backup.json

Configuration

# Show current configuration
langlands config show

# Set configuration values
langlands config set neural.learning_rate 0.001
langlands config set visualization.default_resolution 1920x1080

# Reset to defaults
langlands config reset

Interactive REPL

The REPL provides an interactive environment for mathematical exploration:

langlands> create group g GL 3
Created group g: GL(3)

langlands> create form f g 2
Created automorphic form f: Eisenstein series of weight 2

langlands> compute correspondence
Langlands correspondence: computed ✓
Verified: ✓

langlands> compute hecke 5
T_5(f) = 2.236068

langlands> plot hecke
Plot opened in viewer

langlands> verify ramanujan
Ramanujan conjecture at p=2: ✓

langlands> save session.json
Session saved to: session.json

REPL Commands

• help - Show help message
• vars - List all variables
• create <type> <name> [args] - Create mathematical objects
• compute <operation> - Perform computations
• plot <type> - Generate plots
• verify <property> - Verify properties
• save/load <file> - Session management

Configuration

Configuration is stored in ~/.config/langlands-cli/config.toml:

default_precision = 64
max_iterations = 10000
convergence_threshold = 1e-10

[computation]
enable_parallel = true
enable_gpu = false
cache_results = true

[visualization]
default_resolution = [800, 600]
color_scheme = "viridis"
enable_latex = true

[neural]
default_architecture = "langlands_v1"
learning_rate = 0.001
batch_size = 32

[repl]
history_size = 1000
auto_save = true
prompt = "langlands> "

Output Formats

JSON

{
  "computation": {
    "type": "correspondence",
    "verified": true,
    "details": "..."
  },
  "metadata": {
    "version": "0.1.0",
    "timestamp": "2024-01-01T00:00:00Z"
  }
}

LaTeX

\documentclass{article}
\begin{document}
\section{Langlands Correspondence}
The correspondence between automorphic forms and Galois representations...
\begin{align}
L(s) &= 1.234567
\end{align}
\end{document}

Mathematica

correspondence = {
  "type" -> "Langlands",
  "verified" -> True,
  "lFunction" -> LFunction[s]
};

Examples

Research Workflow

# 1. Set up computation
langlands config set computation.enable_parallel true
langlands config set neural.batch_size 64

# 2. Verify correspondences for GL(n)
for n in 2 3 4; do
  langlands compute correspondence --input "GL($n)" --output "gl${n}_results.json"
done

# 3. Train neural network on patterns
langlands train --dataset combined_data.json --architecture deep --epochs 200

# 4. Generate visualizations
langlands visual correspondence --output correspondence_diagram.svg
langlands visual hecke-eigenvalues --resolution 2560x1440 --output eigenvalues.png

# 5. Export for publication
langlands export recent --format latex --metadata --output paper.tex

Batch Verification

#!/bin/bash
# Batch verify multiple properties

properties=("correspondence" "functoriality" "ramanujan" "reciprocity")

for prop in "${properties[@]}"; do
  echo "Verifying $prop..."
  langlands verify "$prop" --depth standard --proof > "${prop}_verification.txt"
done

# Generate summary report
langlands export all --format json --output verification_summary.json

Performance

The CLI is optimized for mathematical computations:

• Parallel processing with configurable thread count
• GPU acceleration for supported operations (CUDA)
• Intelligent caching to avoid redundant computations
• Memory-efficient algorithms for large-scale problems
• Progress tracking for long-running computations

Contributing

We welcome contributions! Please see the main repository's contributing guidelines.

License

MIT License - see LICENSE file for details.

Citation

If you use this tool in your research, please cite:

@software{geometric_langlands_cli,
  title = {Geometric Langlands CLI},
  author = {ruv-FANN Contributors},
  url = {https://github.com/ruvnet/ruv-FANN},
  version = {0.1.0},
  year = {2024}
}