# Mathematics - Processes Backlog

## Overview

This document outlines the Phase 2 processes backlog for the Mathematics specialization. These processes are designed to automate and standardize key mathematical workflows across pure mathematics, applied mathematics, statistics, and computational mathematics.

## Process Categories

### Category 1: Proof and Theorem Development

#### 1. theorem-proof-verification
**Priority:** High
**Description:** Formal verification of mathematical proofs using proof assistants (Lean, Coq, Isabelle). Includes translating informal proofs into formal systems, checking logical consistency, and identifying gaps in reasoning.
**Key Activities:**
- Parse informal proof into structured format
- Translate to proof assistant syntax
- Verify each logical step
- Generate counterexample searches
- Report verification results and any gaps

#### 2. conjecture-exploration
**Priority:** Medium
**Description:** Systematic exploration and testing of mathematical conjectures through computational experiments. Generates test cases, searches for counterexamples, and identifies patterns supporting or refuting conjectures.
**Key Activities:**
- Formalize conjecture statement
- Generate test cases across parameter space
- Execute computational verification
- Search for counterexamples
- Document findings and patterns

#### 3. proof-writing-assistance
**Priority:** High
**Description:** Assist in writing rigorous mathematical proofs with proper structure, notation, and LaTeX formatting. Ensures logical flow, identifies missing steps, and suggests clearer formulations.
**Key Activities:**
- Review proof structure and logic
- Identify implicit assumptions
- Suggest clearer notation
- Format in LaTeX
- Generate proof outline templates

### Category 2: Numerical Analysis and Computation

#### 4. numerical-stability-analysis
**Priority:** High
**Description:** Analyze numerical algorithms for stability, convergence, and error propagation. Includes condition number analysis, floating-point error bounds, and stability assessment for various input ranges.
**Key Activities:**
- Identify potential numerical instabilities
- Compute condition numbers
- Analyze floating-point error accumulation
- Test boundary cases
- Recommend stable reformulations

#### 5. algorithm-complexity-analysis
**Priority:** Medium
**Description:** Perform rigorous time and space complexity analysis of mathematical algorithms. Includes asymptotic analysis, average-case analysis, and comparison with alternative approaches.
**Key Activities:**
- Derive big-O time complexity
- Analyze space requirements
- Compute average-case complexity
- Compare with alternative algorithms
- Identify optimization opportunities

#### 6. pde-solver-selection
**Priority:** Medium
**Description:** Guide selection of appropriate numerical methods for partial differential equations based on problem characteristics (elliptic, parabolic, hyperbolic), boundary conditions, and accuracy requirements.
**Key Activities:**
- Classify PDE type and characteristics
- Assess boundary condition types
- Recommend discretization methods
- Suggest appropriate solvers
- Estimate computational requirements

#### 7. matrix-computation-optimization
**Priority:** High
**Description:** Optimize matrix computations by selecting appropriate decompositions, exploiting structure (sparsity, symmetry), and leveraging high-performance libraries (BLAS, LAPACK).
**Key Activities:**
- Analyze matrix structure and properties
- Select optimal decomposition method
- Recommend sparse vs dense algorithms
- Interface with HPC libraries
- Benchmark performance

### Category 3: Statistical Analysis

#### 8. experimental-design-planning
**Priority:** High
**Description:** Design rigorous statistical experiments with proper sample size calculations, randomization schemes, and power analysis. Ensures statistical validity before data collection begins.
**Key Activities:**
- Define research hypothesis
- Calculate required sample size
- Design randomization protocol
- Conduct power analysis
- Document design decisions

#### 9. statistical-model-selection
**Priority:** High
**Description:** Guide selection of appropriate statistical models based on data characteristics, assumptions, and research questions. Includes model comparison using information criteria and cross-validation.
**Key Activities:**
- Assess data distribution and structure
- Check model assumptions
- Compare candidate models (AIC, BIC, CV)
- Validate model diagnostics
- Document selection rationale

#### 10. bayesian-inference-workflow
**Priority:** Medium
**Description:** Implement Bayesian inference workflows including prior elicitation, posterior computation via MCMC/variational methods, and convergence diagnostics.
**Key Activities:**
- Specify prior distributions
- Implement likelihood function
- Run MCMC sampling (Stan, PyMC)
- Check convergence diagnostics
- Generate posterior summaries

#### 11. hypothesis-testing-framework
**Priority:** Medium
**Description:** Conduct rigorous hypothesis testing with proper multiple comparison corrections, effect size estimation, and confidence interval construction.
**Key Activities:**
- Formalize null and alternative hypotheses
- Select appropriate test statistic
- Apply multiple comparison corrections
- Compute effect sizes
- Report results with uncertainty

### Category 4: Optimization

#### 12. optimization-problem-formulation
**Priority:** High
**Description:** Translate real-world optimization problems into mathematical programming formulations. Identifies decision variables, objective functions, and constraints from problem descriptions.
**Key Activities:**
- Extract decision variables
- Formulate objective function
- Identify and encode constraints
- Classify problem structure
- Generate model in standard form

#### 13. convex-analysis-verification
**Priority:** Medium
**Description:** Verify convexity properties of optimization problems to enable efficient solution methods. Includes checking convexity of objective and constraints, and identifying hidden convex structure.
**Key Activities:**
- Check objective convexity (Hessian analysis)
- Verify constraint set convexity
- Identify reformulations to convex form
- Recommend appropriate solvers
- Document convexity analysis

#### 14. sensitivity-analysis-optimization
**Priority:** Medium
**Description:** Perform sensitivity analysis on optimization solutions to understand parameter impacts, identify binding constraints, and assess solution robustness.
**Key Activities:**
- Compute dual variables
- Analyze binding constraints
- Perform parametric analysis
- Identify critical parameters
- Generate sensitivity reports

### Category 5: Mathematical Modeling

#### 15. model-formulation-workflow
**Priority:** High
**Description:** Systematic workflow for developing mathematical models from real-world systems. Includes dimensional analysis, assumption documentation, and derivation of governing equations.
**Key Activities:**
- Document system variables and parameters
- State and justify assumptions
- Apply dimensional analysis
- Derive governing equations
- Validate against known limits

#### 16. uncertainty-quantification
**Priority:** High
**Description:** Quantify uncertainty in mathematical models through sensitivity analysis, parameter uncertainty propagation, and Monte Carlo methods.
**Key Activities:**
- Identify uncertain parameters
- Propagate uncertainty through model
- Perform sensitivity analysis
- Generate confidence bounds
- Document uncertainty sources

#### 17. model-validation-framework
**Priority:** Medium
**Description:** Validate mathematical models against experimental or observational data. Includes parameter estimation, residual analysis, and goodness-of-fit assessment.
**Key Activities:**
- Define validation metrics
- Compare model predictions to data
- Perform residual analysis
- Assess predictive accuracy
- Document model limitations

### Category 6: Symbolic Computation

#### 18. symbolic-simplification
**Priority:** Medium
**Description:** Simplify complex mathematical expressions using computer algebra systems. Includes algebraic simplification, trigonometric identities, and pattern recognition.
**Key Activities:**
- Parse mathematical expression
- Apply simplification rules
- Identify patterns and structures
- Generate equivalent forms
- Verify simplification correctness

#### 19. symbolic-integration-differentiation
**Priority:** Medium
**Description:** Perform symbolic integration and differentiation with verification. Handles special functions, definite integrals, and generates step-by-step solutions.
**Key Activities:**
- Parse integral/derivative expression
- Apply symbolic computation
- Verify result correctness
- Handle special cases
- Generate step-by-step solution

### Category 7: Research and Documentation

#### 20. mathematical-literature-review
**Priority:** Medium
**Description:** Conduct systematic literature review for mathematical research. Searches arXiv, MathSciNet, and other databases to identify relevant prior work and open problems.
**Key Activities:**
- Formulate search queries
- Search mathematical databases
- Categorize relevant papers
- Identify gaps and open problems
- Generate annotated bibliography

#### 21. latex-document-generation
**Priority:** High
**Description:** Generate properly formatted LaTeX documents for mathematical content. Includes theorem environments, equation formatting, and bibliography management.
**Key Activities:**
- Structure document layout
- Format mathematical expressions
- Set up theorem environments
- Manage bibliography
- Ensure consistent notation

#### 22. mathematical-notation-standardization
**Priority:** Low
**Description:** Ensure consistent mathematical notation throughout documents and codebases. Includes notation glossaries and automatic notation checking.
**Key Activities:**
- Define notation conventions
- Check notation consistency
- Generate notation glossary
- Suggest standardized alternatives
- Document notation choices

### Category 8: Computational Reproducibility

#### 23. reproducible-computation-setup
**Priority:** High
**Description:** Set up reproducible computational mathematics environments with version control, dependency management, and random seed documentation.
**Key Activities:**
- Configure version control
- Document dependencies
- Set random seeds
- Create reproducible scripts
- Generate reproducibility reports

#### 24. benchmark-validation
**Priority:** Medium
**Description:** Validate mathematical software implementations against standard benchmark problems and analytical solutions.
**Key Activities:**
- Identify appropriate benchmarks
- Run benchmark tests
- Compare to analytical solutions
- Document accuracy and performance
- Generate validation reports

## Summary

| Category | Process Count | Priority Distribution |
|----------|---------------|----------------------|
| Proof and Theorem Development | 3 | 2 High, 1 Medium |
| Numerical Analysis and Computation | 4 | 2 High, 2 Medium |
| Statistical Analysis | 4 | 2 High, 2 Medium |
| Optimization | 3 | 1 High, 2 Medium |
| Mathematical Modeling | 3 | 2 High, 1 Medium |
| Symbolic Computation | 2 | 2 Medium |
| Research and Documentation | 3 | 1 High, 1 Medium, 1 Low |
| Computational Reproducibility | 2 | 1 High, 1 Medium |

**Total Processes:** 24

**Priority Breakdown:**
- High Priority: 11 processes
- Medium Priority: 12 processes
- Low Priority: 1 process

## Implementation Notes

1. **Dependencies:** Many processes share common infrastructure (LaTeX, numerical libraries, proof assistants)
2. **Integration Points:** Processes should integrate with existing mathematical software ecosystems (Python/SciPy, Julia, R, MATLAB)
3. **Verification:** All computational processes should include validation against known solutions
4. **Documentation:** Each process should generate reproducible documentation of methods and results

## Next Steps

1. Implement high-priority processes first
2. Establish testing framework with mathematical benchmarks
3. Create integration with proof assistants (Lean, Coq)
4. Develop templates for common mathematical workflows
5. Build connections to statistical software (R, Stan, PyMC)
