| Algebraic geometry | Schemes, stacks, derived categories, motives | Mother | Shared ambient world where exactness and cohomology transport facts | Proofs hinging on quantitative estimates for singularities or valuations |
| Algebraic geometry | Minimal model program, birational classification | Mother | Global program with canonical reductions and cone technology | Local analytic bounds via multiplier ideals and explicit discrepancy control |
| Algebraic geometry | Geometric invariant theory, stability | Mother | Linearization by line bundles and one parameter subgroup weights | Metric stability links and coercive energy functionals |
| Algebraic geometry | Hodge theory, period maps | Mother | Linear data on  with Lefschetz and Hodge Riemann relations | Variations with curvature estimates and  techniques |
| Arithmetic geometry | Ell adic cohomology, p adic Hodge theory, perfectoid spaces | Mother | Unified comparison and finiteness in a common grammar | Quantitative exponential sum bounds or local monodromy estimates |
| K theory | Higher K theory, Chern character | Mother | Forcing additivity and landing in cohomology | Explicit K theory of singularities with analytic control |
| Category theory | Higher categories, topos theory | Mother | Enlarged container that makes constructions natural | Concrete counterexamples that stress smallness or exactness |
| Homotopy theory | Stable homotopy, spectra | Mother | Powerful shared machinery and exact triangles | Unstable problems that need hard geometric input |
| Representation theory | Geometric representation theory, Langlands categorification | Mother | Proofs flow inside a large functorial edifice | Analytic bounds for automorphic forms and trace formula estimates |
| Topology | Algebraic topology, spectral sequences | Mother | Linear invariants and functorial tools dominate | Metric and curvature inputs in geometric topology |
| Differential geometry | Ricci flow, mean curvature flow | Father | Nonlinear parabolic estimates, monotonicity, and surgery are decisive | Index theory and canonical theorems as scaffolding |
| Differential geometry | Kähler geometry, canonical metrics | Father | Complex Monge Ampere, pluripotential estimates, variational energies | Stability criteria and algebro geometric predictions |
| Geometric analysis | Minimal surfaces, calculus of variations, regularity | Father | Epsilon regularity, blow ups, improvement of flatness | Large regularity frameworks that centralize technique |
| Microlocal and semiclassical analysis | Pseudo differential calculus, parametrices | Mixed leaning Mother | A calculus that centralizes detail and shortens later proofs | Closing parametrix errors by sharp local inequalities |
| Harmonic analysis | Calderon Zygmund, time frequency, decoupling | Mixed leaning Father | Local inequalities, extrapolation, scale analysis with visible losses | Library theorems once the right function space is fixed |
| Functional analysis | Operator algebras, Banach space theory | Mixed leaning Mother | Abstract structure, exactness, duality | Quantitative geometry of Banach spaces and sharp type cotype estimates |
| Spectral theory | Scattering, resonances, eigenvalue asymptotics | Mixed | Abstract self adjoint theory is Mother | Local resolvent bounds and Carleman style estimates pull to Father |
| Partial differential equations | Elliptic and parabolic regularity | Father | Caccioppoli, Schauder, De Giorgi, Moser on page estimates | Compactness theorems as background only |
| Partial differential equations | Dispersive and wave equations | Father | Strichartz, Morawetz, profile decompositions, rigidity arguments | Abstract functional calculus as secondary support |
| Fluid mechanics | Euler, Navier Stokes, boundary layers | Father | Supercritical regimes and vortex dynamics require bespoke control | Weak compactness frameworks as partial support |
| Free boundary problems | Obstacle, Bernoulli, Stefan | Father | Monotonicity formulas, blow ups, epiperimetric inequalities | Variational reformulations after estimates are proved |
| Kinetic theory | Boltzmann, Landau, Vlasov | Father | Hypocoercivity and weighted energy couple derivatives and moments | Spectral gap theory helps but closure is analytic |
| Inverse problems and control | Carleman estimates, observability | Father | Geometry matched weights and sign budgets decide unique continuation | Abstract semigroup language as scaffolding |
| Stochastic PDE | Regularity structures, paracontrolled calculus | Father | Scale by scale renormalization and fixed points under low regularity | The framework hosts but does not replace the estimates |
| Probability | Interacting particle systems, renormalization | Father | Sharp thresholds and multi scale flow with error budgets | Abstract Gibbsian formalism as background only |
| Probability | Ergodic theory and dynamics | Mixed | Structure theorems and factors are Mother | Quantitative mixing and rigidity are Father |
| Probability | Random matrix theory | Mixed leaning Father | Local laws, universality, rigidity via inequalities | Global semicircle and free probabilistic limits lean Mother |
| Probability | Abstract measure theory | Mother | Containment and transfer dominate | Sharp functional inequalities tilt to Father |
| Analytic number theory | Primes and gaps, sieves, exponent pairs, L functions bounds | Father | Cancellation and explicit loss absorption at critical scales | Trace formula and automorphic representation theory bring Mother edifice |
| Additive combinatorics | Structure versus randomness, quantitative Szemeredi | Mixed | Decomposition is Mother; quantitative bounds and transference are Father | Choice of target estimate sets the balance |
| Automorphic forms and Langlands | Trace formula, endoscopy, functoriality | Mother | Global architecture and transport across groups | Local harmonic analytic estimates as Father steps |
| Arithmetic geometry | Diophantine geometry, heights, Arakelov | Mixed leaning Mother | Unified formalism and heights as linear data | Effective bounds and equidistribution estimates |
| Information theory | Shannon coding and asymptotics | Mother | Unified axioms and convex dualities | Finite blocklength and nonasymptotic bounds pull to Father |
| Coding theory | Algebraic geometric codes, list decoding | Mother | Algebraic constructions travel by dictionary | Finite length reliability and performance analysis |
| Cryptography | Lattice and algebraic constructions | Mixed leaning Mother | Security reductions in a shared model | Concrete side channel and parameter attacks as Father testing |
| Computational number theory | Lattice algorithms, primality, factorization | Mixed | Complexity bounds use both edifice and craft | Hard average case analysis pulls to Father |
| Algorithms and optimization | Linear and semidefinite programming, rounding | Mother | Convexity and hierarchies gather many instances under one grammar | Integrality gaps and forbidden minors force Father limits |
| Algorithms and optimization | Nonconvex optimization, landscape analysis | Father | Geometry of critical points and escape rates are quantitative | When convex relaxations exist the tilt moves toward Mother |
| Streaming and communication | Sketches and lower bounds | Father | Information budgets and adversarial adaptivity | Algebraic sketches as small Mother engines inside a Father proof |
| Data structures | Cell probe, chronogram lower bounds | Father | Operation pattern invariants decide costs | Only high level model adds Mother language |
| Online and learning with adversaries | Regret and potential methods | Father | Calibrated budgets against adaptive feedback | Benign statistical settings tilt to Mother |
| Statistical learning theory | PAC, structural risk, VC theory | Mixed leaning Mother | Global generalization frames | Adversarial or adaptive settings flip to Father |
| Numerical analysis | Stability, convergence, error control | Father | Mesh and time step budgets and energy arguments are explicit | Abstract stability from functional analysis as support |
| Control theory | Optimal control, PDE control | Father | Carleman weights, reachability, and costs are quantitative | Linear systems lore adds Mother language |
| Mathematical physics | Constructive QFT, rigorous renormalization group | Father | Scale by scale control with renormalization | Axiomatic schemes and categories as Mother frames |
| Mathematical physics | TQFT, conformal field theory, categorical formulations | Mother | Functoriality and tensor categories centralize detail | Analytic boundary conformal estimates pull to Father |
| Symplectic and contact | Floer theory and holomorphic curves | Mixed | Analytic transversality and bubbling control are Father | Categorical and homological packages are Mother |
| Low dimensional topology | Gauge theory, Seiberg Witten, Donaldson | Father | Nonlinear elliptic estimates and compactness | Once established, the framework acts as a Mother engine |
| Operator algebras | C star and von Neumann algebras | Mother | Structural classification and exactness properties | Quantitative noncommutative ergodic theorems lean Father |
| Graph theory | Extremal and probabilistic graph theory | Father | Inequalities, thresholds, containers, nibble methods | Graph limits and regularity give Mother devices |
| Graph theory | Graph minors and structure theory | Mother | Global decomposition with certified models | Algorithmic hardness proofs add Father components |
| Combinatorics | Enumerative methods and generating functions | Mother | Algebraic formalism transports identities | Asymptotics by saddle point or steepest descent invite Father estimates |
| Geometry processing and applied geometry | Discrete differential geometry | Mixed leaning Father | Stability and error control for numerical schemes | Library solvers and convex relaxations move toward Mother |
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