# Math Expert Configuration # 1.1. Mathematical Domains and Specializations MATH_DOMAINS = { "algebra": { "level": "expert", "topics": [ "linear algebra", "abstract algebra", "polynomial equations", "matrix operations", "group theory", "ring theory", "field theory", "representation theory", "homological algebra", "category theory", "universal algebra", "non-associative algebras", "Lie algebras", "quantum groups", "Hopf algebras", "K-theory" ] }, "calculus": { "level": "expert", "topics": [ "single variable calculus", "multivariable calculus", "differential equations", "partial differential equations", "vector calculus", "complex analysis", "functional analysis", "measure theory", "differential geometry", "geometric measure theory", "non-standard analysis", "stochastic calculus", "calculus of variations", "symplectic geometry" ] }, "proof_writing": { "level": "expert", "topics": [ "induction", "contradiction", "direct proof", "proof by cases", "epsilon-delta proofs", "existence proofs", "uniqueness proofs", "category theory proofs", "homotopy type theory", "model theory", "proof theory", "set theory", "constructive mathematics", "proof complexity" ] }, "probability": { "level": "expert", "topics": [ "probability theory", "random variables", "distributions", "stochastic processes", "Bayesian inference", "Markov chains", "measure-theoretic probability", "stochastic calculus", "martingales", "large deviations", "ergodic theory", "random matrix theory", "stochastic PDEs" ] }, "statistics": { "level": "expert", "topics": [ "descriptive statistics", "inferential statistics", "hypothesis testing", "regression analysis", "time series analysis", "bayesian statistics", "non-parametric methods", "statistical learning theory", "high-dimensional statistics", "causal inference", "spatial statistics", "robust statistics", "computational statistics" ] }, "number_theory": { "level": "expert", "topics": [ "prime numbers", "modular arithmetic", "diophantine equations", "cryptography", "analytic number theory", "algebraic number theory", "elliptic curves", "automorphic forms", "arithmetic geometry", "p-adic analysis", "analytic continuation", "modular forms", "zeta functions" ] }, "geometry": { "level": "expert", "topics": [ "euclidean geometry", "non-euclidean geometry", "differential geometry", "topology", "algebraic geometry", "projective geometry", "symplectic geometry", "algebraic topology", "geometric analysis", "geometric group theory", "Riemannian geometry", "Kähler geometry", "hyperbolic geometry" ] }, "combinatorics": { "level": "expert", "topics": [ "graph theory", "enumerative combinatorics", "combinatorial optimization", "matroid theory", "combinatorial designs", "extremal combinatorics", "probabilistic combinatorics", "algebraic combinatorics", "topological combinatorics", "combinatorial geometry", "Ramsey theory" ] }, "logic": { "level": "expert", "topics": [ "first-order logic", "model theory", "proof theory", "set theory", "computability theory", "type theory", "category theory", "modal logic", "temporal logic", "constructive logic", "intuitionistic logic", "proof complexity" ] }, "theoretical_cs": { "level": "expert", "topics": [ "computational complexity", "algorithms", "cryptography", "quantum computing", "machine learning theory", "formal verification", "type systems", "programming language theory", "distributed computing", "parallel algorithms", "computational geometry", "randomized algorithms" ] }, "applied_math": { "level": "expert", "topics": [ "numerical analysis", "optimization", "control theory", "mathematical physics", "fluid dynamics", "quantum mechanics", "relativity", "mathematical biology", "financial mathematics", "signal processing", "data assimilation", "inverse problems" ] } } # 1.2. Core Tasks CORE_TASKS = [ { "task_type": "problem_solving", "description": "Solve complex mathematical problems", "example": "Prove the Riemann Hypothesis", "difficulty_levels": ["basic", "intermediate", "advanced", "research_level", "open_problem"] }, { "task_type": "proof_writing", "description": "Prove mathematical statements with advanced techniques", "example": "Prove Fermat's Last Theorem using elliptic curves", "proof_types": ["induction", "contradiction", "direct", "cases", "category_theory", "homotopy_type", "model_theory", "proof_complexity", "constructive"] }, { "task_type": "calculus_computation", "description": "Perform advanced calculus operations", "example": "Solve Navier-Stokes equations for turbulence", "operation_types": ["differentiation", "integration", "limits", "functional_analysis", "measure_theory", "stochastic_calculus", "geometric_measure_theory"] }, { "task_type": "symbolic_computation", "description": "Manipulate complex mathematical expressions", "example": "Simplify tensor equations in general relativity", "expression_types": ["polynomial", "rational", "trigonometric", "exponential", "tensor", "operator", "Lie_algebra", "Hopf_algebra"] }, { "task_type": "concept_explanation", "description": "Explain advanced mathematical concepts", "example": "Explain the Langlands program", "explanation_types": ["definition", "intuition", "application", "example", "formal", "geometric", "historical", "pedagogical"] }, { "task_type": "statistical_analysis", "description": "Perform advanced statistical analysis", "example": "Analyze high-dimensional genomic data", "statistical_methods": ["regression", "hypothesis_testing", "confidence_intervals", "bayesian_methods", "non_parametric", "causal_inference", "computational_methods"] }, { "task_type": "probability_calculation", "description": "Calculate complex probabilities", "example": "Calculate phase transitions in random matrix theory", "distributions": ["binomial", "normal", "poisson", "exponential", "multivariate", "stochastic_processes", "random_matrix", "levy_processes"] }, { "task_type": "number_theory_problem", "description": "Solve advanced number theory problems", "example": "Prove the Birch and Swinnerton-Dyer conjecture", "problem_types": ["prime", "modular", "diophantine", "analytic", "algebraic", "elliptic_curve", "modular_form"] }, { "task_type": "geometric_construction", "description": "Construct and analyze complex geometric objects", "example": "Construct a Calabi-Yau manifold", "construction_types": ["euclidean", "non_euclidean", "projective", "differential", "algebraic", "symplectic", "topological"] }, { "task_type": "mathematical_modeling", "description": "Create advanced mathematical models", "example": "Model quantum field theory", "model_types": ["continuous", "discrete", "stochastic", "partial_differential", "non_linear", "quantum", "statistical"] }, { "task_type": "proof_verification", "description": "Verify complex mathematical proofs", "example": "Verify the proof of the Four Color Theorem", "verification_methods": ["formal_verification", "model_checking", "proof_assistant", "automated_reasoning", "interactive_theorem_proving"] }, { "task_type": "algorithm_design", "description": "Design and analyze mathematical algorithms", "example": "Design a quantum algorithm for factorization", "algorithm_types": ["numerical", "combinatorial", "geometric", "algebraic", "probabilistic", "quantum", "parallel"] }, { "task_type": "research_paper_analysis", "description": "Analyze and explain mathematical research papers", "example": "Explain Wiles' proof of Fermat's Last Theorem", "analysis_types": ["technical", "historical", "pedagogical", "critical", "extensional"] }, { "task_type": "open_problem_analysis", "description": "Analyze and make progress on open mathematical problems", "example": "Analyze the Collatz conjecture", "problem_classes": ["number_theory", "combinatorics", "analysis", "algebra", "geometry", "probability"] }, { "task_type": "mathematical_philosophy", "description": "Analyze philosophical aspects of mathematics", "example": "Explain the foundations of mathematics", "philosophical_topics": ["foundations", "philosophy_of_math", "logic", "set_theory", "constructivism", "intuitionism"] }, { "task_type": "mathematical_software_development", "description": "Develop mathematical software and algorithms", "example": "Implement a new numerical method", "software_types": ["numerical", "symbolic", "proof_assistant", "visualization", "simulation", "optimization"] } ] # Dataset Configuration DATASETS = { "proofnet": { "source": "huggingface", "dataset_name": "proofnet", "split": "train", "use_fields": ["problem", "solution", "proof_steps"] }, "math_dataset": { "source": "huggingface", "dataset_name": "deepmind/mathematics_dataset", "split": "train-hard", "use_fields": ["question", "answer", "steps"] }, "gsm8k": { "source": "huggingface", "dataset_name": "gsm8k", "split": "train", "use_fields": ["question", "answer"] }, "mathlib": { "source": "huggingface", "dataset_name": "mathlib", "split": "train", "use_fields": ["theorem", "proof", "dependencies"] }, "arxiv_math": { "source": "huggingface", "dataset_name": "arxiv_math", "split": "train", "use_fields": ["paper", "equations", "proofs"] }, "clay_institute": { "source": "huggingface", "dataset_name": "clay_institute_problems", "split": "train", "use_fields": ["problem", "background", "current_status", "approaches"] }, "open_problems": { "source": "huggingface", "dataset_name": "open_math_problems", "split": "train", "use_fields": ["problem", "category", "history", "attempts"] }, "research_papers": { "source": "huggingface", "dataset_name": "math_research_papers", "split": "train", "use_fields": ["title", "abstract", "content", "proofs", "theorems"] } } # Data Processing Configuration DATA_PROCESSING = { "format": "jsonl", "normalization": { "equations": "sympy", "latex": "plaintext", "proof_steps": "yaml", "tensor_operations": "torch", "quantum_operations": "qiskit", "geometric_objects": "geometric_algebra", "category_theory": "category_theory" }, "validation": { "min_steps": 2, "max_steps": 200, "min_length": 10, "max_length": 100000 } } if __name__ == "__main__": print("Math Expert Configuration Loaded") print(f"Number of domains: {len(MATH_DOMAINS)}") print(f"Number of tasks: {len(CORE_TASKS)}") print(f"Number of datasets: {len(DATASETS)}")