Mathematics skills for AI agents

Triggers when users need help with abstract algebra. Activate for questions about groups, rings, fields, homomorphisms, isomorphisms, quotient groups, quotient rings, Sylow theorems, normal subgroups, cosets, Lagrange's theorem, polynomial rings, ideals, field extensions, Galois theory, group actions, and applications in cryptography and coding theory.

Triggers when users need help with calculus concepts and problems. Activate for questions about limits, continuity, derivatives, integrals, series, Taylor expansions, Fourier series, vector calculus, gradient, divergence, curl, Stokes' theorem, Green's theorem, multivariable calculus, partial derivatives, chain rule, implicit differentiation, integration by parts, substitution, improper integrals, and applications of calculus in physics and engineering.

Triggers when users need help with complex analysis. Activate for questions about complex functions, analytic functions, holomorphic functions, Cauchy-Riemann equations, contour integration, Cauchy's integral theorem, Cauchy's integral formula, Laurent series, residue theorem, poles, singularities, conformal mappings, Mobius transformations, and applications in fluid dynamics and electrical engineering.

Triggers when users need help with differential equations. Activate for questions about ODEs, PDEs, separable equations, exact equations, linear ODEs, Bernoulli equations, systems of ODEs, heat equation, wave equation, Laplace equation, Fourier series, Fourier transforms, Laplace transforms, numerical methods for ODEs, Euler method, Runge-Kutta, stability analysis, phase portraits, and boundary value problems.

Triggers when users need help with discrete mathematics. Activate for questions about combinatorics, permutations, combinations, pigeonhole principle, inclusion-exclusion, graph theory, paths, trees, graph coloring, planar graphs, network flow, recurrence relations, generating functions, Boolean algebra, lattices, partial orders, and applications of discrete math in computer science.

Triggers when users need help with geometry or trigonometry. Activate for questions about Euclidean geometry, coordinate geometry, geometric transformations, rotation, reflection, scaling, trigonometric functions, trigonometric identities, spherical geometry, projective geometry, differential geometry, curves, surfaces, curvature, geodesics, computational geometry algorithms, convex hulls, and Voronoi diagrams.

Triggers when users need help with linear algebra. Activate for questions about vectors, matrices, vector spaces, linear transformations, eigenvalues, eigenvectors, determinants, rank, null space, SVD, LU decomposition, QR decomposition, orthogonality, inner product spaces, least squares, matrix factorizations, and applications in machine learning, computer graphics, and physics.

Triggers when users need help with mathematical logic and foundations. Activate for questions about propositional logic, predicate logic, proof techniques, direct proof, proof by contradiction, mathematical induction, contrapositive proofs, set theory, ZFC axioms, completeness theorem, incompleteness theorems, Godel, computability, decidability, model theory, formal verification, and automated theorem proving.

Triggers when users need help with mathematical modeling. Activate for questions about model formulation, dimensional analysis, scaling, population models, logistic growth, Lotka-Volterra predator-prey, diffusion models, network models, Monte Carlo simulation, sensitivity analysis, model validation, parameter estimation, compartmental models, epidemiological models, and communicating mathematical results.

Triggers when users need help with number theory. Activate for questions about divisibility, prime numbers, fundamental theorem of arithmetic, modular arithmetic, congruences, Euler's theorem, Fermat's little theorem, quadratic reciprocity, continued fractions, Diophantine equations, primality testing, RSA encryption, elliptic curve cryptography, the Chinese remainder theorem, and number-theoretic algorithms.

Triggers when users need help with numerical methods and scientific computing. Activate for questions about root finding, Newton's method, bisection method, secant method, interpolation, Lagrange interpolation, splines, numerical integration, Simpson's rule, Gaussian quadrature, numerical linear algebra, iterative solvers, condition numbers, numerical ODE solvers, numerical PDE solvers, floating-point arithmetic, error analysis, and numerical stability.

Triggers when users need help with mathematical optimization. Activate for questions about linear programming, simplex method, interior point methods, convex optimization, gradient descent, Newton's method for optimization, proximal methods, constrained optimization, KKT conditions, Lagrange multipliers, integer programming, combinatorial optimization, metaheuristics, genetic algorithms, simulated annealing, and operations research.

Triggers when users need help with probability theory and stochastic processes. Activate for questions about probability spaces, conditional probability, Bayes' theorem, random variables, probability distributions, normal distribution, Poisson distribution, exponential distribution, expectation, variance, moment generating functions, law of large numbers, central limit theorem, Markov chains, martingales, and stochastic modeling.

Triggers when users need help with real analysis. Activate for questions about sequences, series of functions, metric spaces, compactness, connectedness, continuity, uniform continuity, pointwise and uniform convergence, differentiation theory, Riemann integration, Lebesgue integration, measure theory, sigma-algebras, function spaces, dominated convergence theorem, monotone convergence theorem, Fatou's lemma, and Lp spaces.

Triggers when users need help with topology. Activate for questions about topological spaces, open sets, closed sets, compactness, connectedness, Hausdorff spaces, continuous maps, homeomorphisms, homotopy, fundamental group, covering spaces, manifolds, point-set topology, algebraic topology, quotient topology, product topology, topological data analysis, and persistent homology.