Database

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.

Triggers when users need help with Bayesian inference, prior selection, posterior computation, or probabilistic modeling. Activate for questions about prior distributions, conjugate priors, Bayesian vs frequentist approaches, MCMC methods, Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo, Bayesian model comparison, hierarchical models, Stan, PyMC, and probabilistic programming.

Triggers when users need help establishing causal relationships from data, whether experimental or observational. Activate for questions about potential outcomes, Rubin causal model, DAGs, Pearl's framework, propensity scores, instrumental variables, difference-in-differences, regression discontinuity, synthetic control, mediation analysis, and sensitivity analysis for unmeasured confounding.

Triggers when users need help summarizing, describing, or exploring data distributions. Activate for questions about mean, median, mode, variance, standard deviation, IQR, skewness, kurtosis, five-number summary, box plots, histograms, outlier detection, data summarization, and exploratory data analysis techniques.

Triggers when users need help designing experiments, clinical trials, or A/B tests. Activate for questions about randomized controlled trials, factorial designs, block designs, Latin square, split-plot designs, A/B testing methodology, power analysis, randomization, blinding, confounding, observational study design, and quasi-experimental methods.

Triggers when users need help with hypothesis testing, confidence intervals, or drawing conclusions from sample data. Activate for questions about null and alternative hypotheses, p-values, Type I and II errors, statistical power, t-tests, ANOVA, chi-square tests, effect sizes, multiple comparisons, Bonferroni correction, FDR, and sample size determination.

Triggers when users need help analyzing datasets with multiple variables simultaneously. Activate for questions about PCA, principal component analysis, factor analysis, canonical correlation, MANOVA, discriminant analysis, cluster analysis, k-means, hierarchical clustering, DBSCAN, multidimensional scaling, correspondence analysis, and structural equation modeling.

Triggers when users need help with distribution-free statistical methods or robust alternatives to parametric tests. Activate for questions about Mann-Whitney U, Wilcoxon signed-rank, Kruskal-Wallis, Spearman correlation, kernel density estimation, bootstrapping, permutation tests, rank-based methods, when to use nonparametric vs parametric tests, and robust statistics.

Triggers when users need help with regression modeling, prediction, or understanding relationships between variables. Activate for questions about linear regression, multiple regression, polynomial regression, logistic regression, generalized linear models, Ridge, Lasso, Elastic Net regularization, residual analysis, multicollinearity, heteroscedasticity, variable selection, and interaction effects.

Triggers when users need help analyzing geographically referenced data or spatial patterns. Activate for questions about spatial autocorrelation, Moran's I, geostatistics, kriging, variograms, point process models, spatial regression, areal data analysis, GIS integration, spatial clustering, hot spot analysis, and applications in epidemiology, ecology, and urban planning.

Triggers when users need help designing surveys, selecting samples, or analyzing complex survey data. Activate for questions about simple random sampling, stratified sampling, cluster sampling, multistage sampling, systematic sampling, survey design, weighting, calibration, non-response adjustment, variance estimation for complex surveys, and questionnaire design principles.

Triggers when users need help analyzing time-to-event data, censored observations, or hazard modeling. Activate for questions about censoring, Kaplan-Meier estimator, log-rank test, Cox proportional hazards model, accelerated failure time models, competing risks, time-varying covariates, survival curves, reliability analysis, and customer churn modeling.

Triggers when users need help with temporal data, forecasting, or time-dependent modeling. Activate for questions about stationarity, autocorrelation, ARIMA models, seasonal decomposition, exponential smoothing, GARCH, state space models, spectral analysis, change point detection, forecasting accuracy metrics, Prophet, and modern time series approaches.

Activate this skill when the user is working with the foundations of algebraic geometry: zero sets of polynomials, the Zariski topology, the dictionary between ideals and varieties, projective closures, dimension and singular points. Triggers on "affine variety," "projective variety," "Zariski topology," "Nullstellensatz," "homogenization," "projective closure," "irreducible components," "coordinate ring," "Jacobian criterion," "singular point," "twisted cubic," "smooth conic," or "algebraic geometry" foundations. Covers the ideal-variety correspondence, dimension theory, smoothness, and worked examples on the twisted cubic, conics and elliptic curves, each verified in Macaulay2, Singular or SageMath.

Activate this skill when the user is working with algebraic curves: genus, divisors, linear systems, the Riemann-Roch theorem and its consequences, elliptic curves and their group law, hyperelliptic curves, ramified covers and embeddings into projective space. Triggers on "algebraic curve," "Riemann-Roch," "genus," "divisor," "linear system," "canonical divisor," "canonical embedding," "hyperelliptic," "elliptic curve," "group law," "Weierstrass form," "Hurwitz formula," "Riemann-Hurwitz," "ramification," "degree-genus formula," "very ample," or curves in "algebraic geometry." Covers precise statements, worked Riemann-Roch computations for genus 0 through 3, the derivation of Weierstrass form and the group law, and how to check each computation in SageMath, Macaulay2 or Singular.

Activate this skill when the user has an algebraic geometry problem to attack rather than a definition to look up: a qualifying exam question, a lemma a paper needs, a conjecture to test, a proof a referee has questioned. Triggers on "how do I prove," "dimension count," "incidence correspondence," "general member," "generic," "Bertini," "semicontinuity," "local coordinates," "affine chart," "compute an example," "Gröbner check," "reduce mod p," "specialization," "degeneration," "standard tricks," "write the proof," "referee," "algebraic geometry exercise," or "algebraic geometry" problem solving. Covers the order of attack, dimension counting done rigorously, local coordinates and charts, choosing the right cohomological tool, what a computation proves and what it does not, how to write an argument that survives refereeing, and a catalogue of the tricks working geometers reach for first.

Activate this skill when the user needs the commutative algebra that algebraic geometry actually runs on, with every theorem tied to its geometric meaning. Triggers on "Noetherian ring," "localization," "local ring," "integral extension," "integral closure," "normalization," "Noether normalization," "Krull dimension," "primary decomposition," "associated primes," "embedded component," "regular local ring," "Cohen-Macaulay," "Hilbert polynomial," "Hilbert series," "depth," or requests to translate between ring theory and geometry. Covers the theorems, what each one says about a variety, and how to compute each invariant in Macaulay2, Singular and SageMath.

Activate this skill when the user wants to compute with polynomial ideals: test membership, eliminate variables, solve polynomial systems, implicitize a parametrization, or understand why a computation will not finish. Triggers on "Gröbner basis," "Groebner basis," "monomial order," "lex order," "grevlex," "Buchberger algorithm," "S-polynomial," "division algorithm," "elimination ideal," "implicitization," "solving polynomial systems," "zero-dimensional ideal," "Macaulay2," "Singular," "SageMath," or computational "algebraic geometry." Covers the theory precisely, real sessions in Macaulay2, Singular and Sage with actual syntax, and the complexity facts that decide which computations are feasible.

Activate this skill when the user needs to count intersections of subvarieties with the correct multiplicities, work with divisors and intersection numbers on surfaces, or reason about blow-ups and exceptional curves. Triggers on "Bézout's theorem," "Bezout," "intersection multiplicity," "intersection number," "Chow group," "Chow ring," "self-intersection," "(-1)-curve," "exceptional divisor," "blow-up," "adjunction formula," "Hodge index," "Riemann-Roch for surfaces," "27 lines," "cubic surface," or intersection-theoretic "algebraic geometry." Covers the local definition of multiplicity, Bézout in the plane and in P^n, the Chow ring of projective space and of products, the intersection pairing on surfaces with blow-ups and adjunction, and worked surface examples checked in Singular, Macaulay2 and Sage.

Activate this skill when the user is learning or using the language of schemes: Spec and Proj, structure sheaves, generic points, nilpotents, fibre products, and the properties of morphisms that carry the geometry. Triggers on "scheme," "Spec," "Proj," "structure sheaf," "generic point," "nilpotent," "fibre product," "closed immersion," "open immersion," "separated," "proper morphism," "finite type," "finite morphism," "flat family," "flat limit," "valuative criterion," or "why schemes" in algebraic geometry. Covers the definitions with their reasons, the standard morphism properties and how to check them, flatness as continuity of fibres, and concrete computations of fibres and flat limits in Macaulay2 and Sage.

Activate this skill when the user needs to work with sheaves on varieties and schemes and to compute or use sheaf cohomology. Triggers on "sheaf," "quasi-coherent," "coherent sheaf," "line bundle," "invertible sheaf," "divisor," "Picard group," "Cech cohomology," "Čech cohomology," "sheaf cohomology," "H^1," "Serre duality," "canonical bundle," "vanishing theorem," "Kodaira vanishing," "Serre vanishing," "Castelnuovo-Mumford regularity," "long exact sequence," "ideal sheaf sequence," "Euler characteristic," "Hilbert polynomial," or cohomological "algebraic geometry." Covers the definitions, the divisor-line bundle dictionary, Čech computations done by hand on projective space, the statement and use of Serre duality, the vanishing theorems that actually get used, Euler characteristic bookkeeping, and how to check every number in Macaulay2, Singular or SageMath.

Activate this skill when the user is building or analysing toric varieties: turning cones, fans and lattice polytopes into varieties, reading smoothness, completeness and projectivity off the fan, computing torus-invariant divisors, their polytopes and sections, or resolving toric singularities by subdivision. Triggers on "toric variety," "fan," "rational polyhedral cone," "lattice polytope," "normal fan," "orbit-cone correspondence," "torus-invariant divisor," "Cox ring," "Hirzebruch surface," "weighted projective space," "Demazure vanishing," "Ehrhart polynomial," "Hilbert basis," "toric ideal," "reflexive polytope," "toric Fano," or combinatorial "algebraic geometry." Covers the cone-fan-polytope dictionary with precise statements, the criteria that decide geometry from combinatorics, divisors and cohomology through lattice points, and working sessions in SageMath, Macaulay2 and Singular.

Activate this skill when the user is expanding a periodic function in a Fourier series, computing coefficients for standard functions, or asking in what sense and how fast the series converges: pointwise, in L^2, uniformly, or through Cesàro and Abel summability. Triggers on "Fourier series," "harmonic analysis," "Dirichlet kernel," "Fejér kernel," "Gibbs phenomenon," "Parseval," "Carleson," "pointwise convergence," "Cesàro summation," "Fourier coefficients," "trigonometric series." Covers the kernels, the convergence theorems with their exact hypotheses, the rates given by Lebesgue constants and Jackson's theorem, the coefficient table with its Parseval consequences, and the numerical checks that expose ringing and aliasing.

Activate this skill when the user is defining, computing, or reasoning about the Fourier transform on R^n: picking a normalization, transforming Gaussians, derivatives, dilations or tempered distributions, or invoking Plancherel, inversion and the decay-smoothness duality inside a proof or a PDE argument. Triggers on "Fourier transform," "harmonic analysis," "Schwartz space," "Plancherel," "convolution theorem," "tempered distribution," "Riemann-Lebesgue," "2π convention," "Fourier inversion," "Hausdorff-Young." Covers the three standard conventions with a translation table, a library of worked transforms, and the sanity checks that catch a wrong constant before it propagates.

Activate this skill when the user is doing Fourier analysis on a group other than R^n: characters and the dual of a locally compact abelian group, Haar measure, the DFT on Z/nZ, Fourier series on the torus, Walsh-Hadamard analysis on the hypercube, Peter-Weyl for compact groups, representation-theoretic Fourier transforms on finite nonabelian groups, or Poisson summation and its consequences. Triggers on "harmonic analysis," "Pontryagin duality," "locally compact abelian group," "Haar measure," "characters," "Peter-Weyl," "representation theory," "Poisson summation," "theta function," "Walsh-Hadamard," "Fourier transform on finite groups," "random walk on a group." Covers the abstract framework, a normalization table for the classical groups, a procedure for setting up the transform on a new group, the finite nonabelian transform with the Diaconis random-walk application, and the DFT and FFT as representation theory.

Activate this skill when the user needs to decompose a function into dyadic frequency pieces, bound a Fourier multiplier on L^p, define or compare Sobolev, Besov and Triebel-Lizorkin spaces, apply Bernstein inequalities, or estimate products and nonlinear terms with paraproducts in a PDE argument. Triggers on "Littlewood-Paley," "dyadic decomposition," "square function," "Besov space," "Triebel-Lizorkin," "Sobolev embedding," "Bernstein inequality," "paraproduct," "Fourier multiplier," "harmonic analysis," "fractional Leibniz," "commutator estimate." Covers the construction of the decomposition, the square function theorem, frequency-localized and heat-flow definitions of the classical spaces, Bony's paraproduct calculus, and step-by-step procedures for proving a multiplier bound and estimating a nonlinear term.

Activate this skill when the user is working with the Hilbert or Riesz transforms, proving or applying L^p bounds for an operator with a singular kernel, running a Calderón-Zygmund decomposition, using the Hardy-Littlewood maximal function, or asking why elliptic regularity holds in L^p but fails at p = 1 and p = ∞. Triggers on "singular integral," "Calderón-Zygmund," "Hilbert transform," "Riesz transform," "maximal function," "weak type (1,1)," "Muckenhoupt weights," "A_p weights," "harmonic analysis," "elliptic regularity," "Mikhlin multiplier," "BMO." Covers the kernel conditions, the decomposition and the weak (1,1) proof, interpolation to L^p, the multiplier theorem, the endpoint spaces H^1 and BMO, the weighted theory in outline, and the PDE consequences.

Activate this skill when the user is estimating a power spectrum from sampled data and needs to choose between the periodogram, Welch averaging and multitaper methods, pick a window, understand leakage and the resolution-variance trade-off, detect sinusoidal lines against noise, or interpret a plotted spectrum without fooling themselves. Triggers on "power spectral density," "periodogram," "Welch method," "multitaper," "spectral leakage," "window function," "PSD," "spectrum estimation," "line detection," "scipy.signal.welch," "harmonic analysis." Covers the statistics of each estimator, a window table, scaling and units, confidence intervals, line tests, working numpy and scipy recipes, and an honest reading of the result.

Activate this skill when the user is expanding a function on the sphere, solving Laplace's equation in spherical coordinates, using the addition theorem or Funk-Hecke formula, choosing normalizations and phase conventions, or computing spherical harmonic transforms for geodesy, graphics lighting or cosmology. Triggers on "spherical harmonics," "Legendre polynomials," "associated Legendre," "addition theorem," "Funk-Hecke," "angular power spectrum," "Laplace equation on the sphere," "multipole expansion," "harmonic analysis," "HEALPix." Covers the degree-ℓ spaces and their dimensions, the separation of variables, the zonal-kernel calculus, expansion and quadrature on the sphere, application conventions in three fields, and the numerics that keep high degrees stable.

Activate this skill when the user is reasoning about how concentrated a function and its Fourier transform can simultaneously be, reconstructing a band-limited signal from samples, diagnosing aliasing, choosing a sampling rate or a window for time-frequency analysis, or citing Heisenberg, Hardy, Donoho-Stark, Shannon-Nyquist or Paley-Wiener correctly. Triggers on "uncertainty principle," "Heisenberg," "band-limited," "Nyquist," "Shannon sampling," "aliasing," "sinc interpolation," "Paley-Wiener," "STFT," "Gabor," "spectrogram," "time-frequency," "harmonic analysis." Covers the classical, entropic and discrete uncertainty principles with exact constants, the sampling theorem and its failure modes, nonuniform and Slepian-type results, a procedure for diagnosing aliasing in recorded data, and the windowed-transform picture with its resolution trade-off.

Activate this skill when the user is building, choosing, or debugging a wavelet decomposition: setting up a multiresolution analysis, deriving scaling and wavelet filters, running the fast wavelet transform, thresholding coefficients for denoising or compression, or dealing with artifacts at signal boundaries. Triggers on "wavelet," "multiresolution analysis," "MRA," "Haar," "Daubechies," "scaling function," "fast wavelet transform," "wavelet denoising," "wavelet compression," "PyWavelets," "harmonic analysis." Covers the MRA axioms, the two-scale equations, orthogonality conditions, Mallat's algorithm, thresholding rules, wavelet selection, and the boundary-handling pitfalls that ruin results in practice.