# zbMATH candidate nodes (raw extractor output — NOT yet deduped)

Format: name | kind | one-line description | dependencies

## CLUSTER A — Number theory / Algebra / Algebraic & arithmetic geometry

1. Large sieve inequality | technique | Inequality bounding mean square of a sequence's exponential sums over well-spaced points or reduced residues, giving near-optimal duality bound. | exponential sums, duality
2. Selberg sieve | technique | Upper-bound sieve minimizing a quadratic form in real weights to bound integers surviving congruence conditions. | quadratic forms, multiplicative structure
3. Parity problem (parity obstruction) | theorem | Selberg's principle that sieve methods alone cannot distinguish integers with even vs odd number of prime factors. | sieve weights, Liouville
4. Bombieri–Vinogradov theorem | theorem | Primes in arithmetic progressions are equidistributed on average over moduli q ≤ x^{1/2-ε}, giving GRH-strength error on average. | large sieve, L-functions
5. GPY sieve (Goldston–Pintz–Yıldırım) | technique | Weighted Selberg-type sieve using truncated divisor-sum weights to detect small gaps between primes. | Selberg sieve, level of distribution
6. Maynard–Tao multidimensional sieve | technique | Higher-dimensional GPY refinement with multivariable weights proving bounded gaps for admissible k-tuples. | GPY sieve, variational problem
7. Zhang's bounded gaps method | technique | Extension of Bombieri–Vinogradov to smooth moduli beyond x^{1/2} via Weil/Deligne bounds, yielding first finite prime-gap bound. | GPY, dispersion, Weil bounds
8. Brun sieve (combinatorial sieve) | technique | Truncated inclusion–exclusion with Brun's pure/upper-lower constructions giving first nontrivial sieve bounds. | inclusion-exclusion
9. Beurling–Selberg extremal functions | technique | Explicit entire band-limited majorants/minorants optimizing the large sieve and explicit-formula error terms. | Fourier analysis, interpolation
10. Hardy–Littlewood circle method | technique | Extraction of asymptotic counts for additive problems by integrating a generating function over major and minor arcs. | exponential sums, arc dissection
11. Kloosterman refinement of the circle method | technique | Circle-method variant summing over full Farey arc using Kloosterman sums to handle forms in fewer variables. | circle method, Kloosterman sums
12. Delta method (Duke–Friedlander–Iwaniec / Heath-Brown) | technique | Circle-method reformulation expressing the detector via a smooth partition over moduli, removing arc dissection. | Poisson summation, Kloosterman sums
13. Singular series | theorem | Arithmetically meaningful product of local densities appearing as the constant in circle-method asymptotics. | major-arc analysis, local densities
14. Weyl differencing | technique | Iterated squaring of an exponential sum to reduce a degree-k polynomial phase to linear, bounding Weyl sums. | Cauchy–Schwarz
15. van der Corput method | technique | Pair of processes (differencing; Poisson/stationary phase) iterated to bound oscillatory sums with smooth phases. | Poisson summation, stationary phase
16. Vinogradov's mean value method | technique | Bounds for solutions of systems of equal-power-sum equations yielding strong Weyl-sum bounds (efficient congruencing / decoupling). | mean value theorems, decoupling
17. Burgess bound | theorem | Nontrivial bound on short character sums of length N ≫ q^{1/4+ε}, beating Pólya–Vinogradov in the short range. | Weil bound, Hölder
18. Pólya–Vinogradov inequality | theorem | Bound of O(√q log q) on partial sums of a primitive character mod q. | Gauss sums
19. Weil bound (Riemann Hypothesis for curves) | theorem | Frobenius eigenvalues on a smooth projective curve over F_q have absolute value q^{1/2}, bounding Kloosterman sums by 2√q. | ℓ-adic cohomology, Frobenius
20. Deligne's bounds (Weil II) | theorem | Frobenius weights on ℓ-adic sheaves are bounded by expected powers of q, giving square-root cancellation for general exponential sums. | étale cohomology, purity
21. Combinatorial Nullstellensatz | theorem | A polynomial nonvanishing in a top coefficient over a large grid forces a nonzero point off a prescribed product set. | polynomial degree counting
22. Croot–Lev–Pach lemma | technique | Rank bound: a low-degree polynomial low-rank as a matrix on a product set controls progression-free set sizes. | polynomial rank, slice rank
23. Slice rank method | technique | Tensor rank notion subadditive on diagonal tensors, bounding combinatorial configurations. | tensor decomposition
24. Ellenberg–Gijswijt cap set bound | theorem | Subsets of F_3^n free of 3-term APs have size O(c^n) with c<3 via polynomial/slice-rank method. | Croot–Lev–Pach, slice rank
25. Stepanov's method | technique | Elementary polynomial construction giving an auxiliary function with many high-order zeros to bound points on curves over finite fields. | auxiliary polynomials, Hasse derivatives
26. Freiman's theorem | theorem | A set of integers with small doubling is contained in a generalized arithmetic progression of bounded rank and size. | Plünnecke–Ruzsa, Fourier
27. Plünnecke–Ruzsa inequality | theorem | Iterated sumset sizes controlled polynomially by the doubling constant. | Ruzsa triangle inequality
28. Balog–Szemerédi–Gowers theorem | theorem | A set with many additive quadruples contains a large subset with small doubling. | graph counting, Plünnecke–Ruzsa
29. Sum-product theorem | theorem | For finite sets in ℝ or F_p, max(|A+A|,|A·A|) ≫ |A|^{1+δ}. | incidence geometry
30. Bohr sets | technique | Approximate-subgroup analogues serving as the locally-abelian-group setting for density-increment arguments. | Fourier analysis, pigeonhole
31. Inverse theorem for Gowers norms (nilsequences) | theorem | A bounded function with large U^{k+1} norm correlates with a degree-k nilsequence. | Gowers norms, nilmanifolds
32. Transference principle | technique | A dense subset of a pseudorandom sparse set inherits additive theorems of the dense integers. | Gowers norms, dense model
33. Density increment argument | technique | Iteratively passing to a subprogression/Bohr set where density strictly increases, terminating in structure. | Fourier L∞ bound, pigeonhole
34. Szemerédi regularity lemma | theorem | Every large graph's vertices partition into a bounded number of pairwise ε-regular parts. | energy increment
35. Arthur–Selberg trace formula | technique | Identity equating a spectral sum over automorphic representations with a geometric sum over conjugacy classes of a reductive group. | adelic harmonic analysis, orbital integrals
36. Theta correspondence (Howe duality) | technique | Representation-theoretic lift between automorphic forms on a reductive dual pair via the Weil representation. | Weil representation, dual pairs
37. Converse theorem | theorem | A Dirichlet series with the right functional equations and twisted continuations is the L-function of an automorphic representation. | Hecke theory, functional equations
38. Cyclic base change | technique | Transfer of automorphic representations of G over F to G over a cyclic extension E/F, matched via the trace formula. | trace formula, twisted endoscopy
39. Automorphy lifting / Taylor–Wiles patching | technique | Proving a residually automorphic Galois representation is automorphic by identifying a deformation ring with a Hecke algebra (R=T). | deformation theory, Taylor–Wiles primes
40. Potential automorphy | technique | Proving a Galois representation becomes automorphic after restriction to a finite extension via Moret-Bailly point-finding plus automorphy lifting. | automorphy lifting, Moret-Bailly
41. Langlands–Shahidi method | technique | Extraction of automorphic L-functions from Fourier coefficients of Eisenstein series' constant/non-constant terms. | Eisenstein series, intertwining operators
42. Geometric Satake equivalence | theorem | Equivalence between perverse sheaves on the affine Grassmannian of G and representations of the Langlands dual group. | affine Grassmannian, perverse sheaves
43. Fundamental lemma (Ngô) | theorem | Identity matching orbital integrals on a group with stable orbital integrals on its endoscopic groups, proved via the Hitchin fibration. | endoscopy, affine Springer fibers
44. Étale cohomology | technique | Weil cohomology theory with ℓ-adic coefficients in the étale topology supporting Galois actions and Lefschetz trace formulas. | Grothendieck topology, Frobenius
45. p-adic Hodge theory | technique | Classification of p-adic Galois representations via Fontaine period rings comparing étale and de Rham/crystalline cohomology. | period rings, (φ,Γ)-modules
46. Perfectoid spaces / tilting | technique | Spaces over perfectoid fields whose tilting equivalence relates char-0 and char-p geometry, with almost purity. | perfectoid fields, almost mathematics
47. Prismatic cohomology | technique | Cohomology over a prism (δ-ring with an ideal) unifying crystalline, de Rham and étale p-adic cohomologies. | δ-rings, derived completion
48. Motivic integration | technique | Integration over arc/jet spaces valued in a Grothendieck ring of varieties computing invariants like stringy cohomology. | arc spaces, Grothendieck ring
49. Chabauty–Coleman method | technique | Bounding rational points on a curve of genus g with Mordell–Weil rank < g via vanishing p-adic Coleman integrals. | p-adic integrals, Jacobian rank
50. Mordell–Weil sieve | technique | Combining reduction maps modulo many primes with the Mordell–Weil group to rule out rational points. | reduction mod p, CRT
51. Theory of heights (height machinery) | technique | Weil height functions and the Néron–Tate canonical height giving a quadratic-form measure of arithmetic complexity. | Weil heights, line bundles
52. Pila–Wilkie theorem / o-minimal point counting | theorem | A definable set in an o-minimal structure has O(H^ε) rational points of height ≤ H off its algebraic part. | o-minimality, Yomdin–Gromov parametrization
53. Zilber–Pink conjecture | state | Atypical intersections of a subvariety with special subvarieties of a Shimura variety lie in finitely many proper special subvarieties. | special subvarieties, o-minimal counting
54. André–Oort theorem | theorem | A subvariety of a Shimura variety with a Zariski-dense set of special (CM) points is itself special. | Pila–Zannier, Galois orbits
55. Faltings's theorem (Mordell conjecture) | theorem | A curve of genus ≥ 2 over a number field has only finitely many rational points. | Arakelov theory, isogeny finiteness
56. Mazur's torsion theorem | theorem | The torsion subgroup of an elliptic curve over ℚ is one of fifteen explicit finite groups. | modular curves, Eisenstein ideal
57. Modularity theorem (Taylor–Wiles–BCDT) | theorem | Every elliptic curve over ℚ is modular, arising from a weight-2 newform. | automorphy lifting, level-lowering
58. Class field theory (Artin reciprocity) | theorem | The abelianized Galois group of a local/global field is described by its idele class group via the Artin map. | idele class group, group cohomology
59. Hasse principle (Brauer–Hasse–Noether) | theorem | A central simple algebra / quadratic form over a number field is split globally iff split everywhere locally. | Brauer group, local invariants
60. Tate–Poitou duality | theorem | Nine-term exact sequence and perfect pairing relating global and local Galois cohomology of a finite module. | Galois cohomology, local duality
61. Euler system method | technique | Compatible family of cohomology classes bounding the size of a Selmer group and an Iwasawa module. | Galois cohomology, Kolyvagin classes
62. Iwasawa Main Conjecture | theorem | The characteristic ideal of an Iwasawa module equals the ideal generated by a p-adic L-function. | Iwasawa algebra, p-adic L-functions
63. Deligne–Lusztig theory | technique | Construction of virtual representations of a finite reductive group in ℓ-adic cohomology of Deligne–Lusztig varieties. | étale cohomology, algebraic groups
64. Springer correspondence | theorem | Bijection relating irreducible Weyl-group representations to nilpotent orbits via cohomology of Springer fibers. | nilpotent cone, perverse sheaves
65. Satake isomorphism | theorem | The spherical Hecke algebra of a p-adic reductive group is isomorphic to the representation ring of the dual group. | unramified representations, dual group
66. Ribet's level-lowering | theorem | A mod-ℓ modular Galois representation arises from a form of lower level under appropriate ramification conditions. | modular Jacobians, congruences
67. Subconvexity / amplification method | technique | Beating the convexity bound for L-functions on the critical line by amplifying a chosen form against a spectral/moment average. | approximate functional equation, moments
68. Affine sieve / property (τ) spectral gap | technique | Using uniform spectral gap for arithmetic quotients to execute sieving in thin/orbit settings. | spectral gap, Bourgain–Gamburd
69. Modular method (Frey curve) | technique | Attaching a Frey elliptic curve to a putative Diophantine solution and deriving a contradiction via modularity + level-lowering. | Frey curve, modularity
70. Baker's method (linear forms in logarithms) | technique | Effective lower bounds for nonzero linear forms in logarithms of algebraic numbers making finiteness results effective. | transcendence theory, zero estimates
71. Stepanov–Bombieri elementary RH for curves | technique | Elementary proof of the Weil bound via Stepanov's auxiliary polynomial avoiding cohomology. | Stepanov's method
72. Hecke algebra method | technique | Study of automorphic/modular forms via the commutative algebra of Hecke double-coset operators. | double cosets, Satake

## CLUSTER B — Analysis / PDE / Harmonic / Probability / Dynamics / Math physics

73. Calderón–Zygmund decomposition | technique | Splits an L¹ function at height λ into a bounded good part and a bad part on disjoint dyadic cubes with zero mean. | dyadic cubes, stopping time
74. Calderón–Zygmund singular integral theory | theorem | Operators with standard kernels bounded on L² extend boundedly to Lᵖ (1<p<∞) and weak-(1,1). | CZ decomposition, interpolation
75. Littlewood–Paley theory | technique | Decomposes a function into dyadic frequency annuli whose square function controls the Lᵖ norm. | Fourier transform, multiplier theorem
76. T(1) theorem (David–Journé) | theorem | A CZ operator is L²-bounded iff T(1), T*(1) ∈ BMO and a weak boundedness property holds. | BMO, Carleson measures
77. T(b) theorem | theorem | Generalizes T(1) by testing against an accretive function b instead of the constant 1. | para-accretive functions, T(1)
78. Stein–Tomas restriction theorem | theorem | The Fourier transform restricts boundedly from Lᵖ(ℝⁿ) to L²(dσ) on the sphere for p ≤ 2(n+1)/(n+3). | stationary phase, TT*
79. Restriction/extension estimates (Stein program) | technique | Bound the extension operator from L^q(sphere) to Lᵖ(ℝⁿ), encoding oscillatory-integral cancellation. | oscillatory integrals, curvature
80. Square function estimates | technique | Control an operator's Lᵖ norm via the L² norm of an associated square function built from frequency or geometric pieces. | Littlewood–Paley, Khintchine
81. Carleson time-frequency / tile analysis | technique | Proves a.e. convergence of Fourier series by organizing phase-plane tiles into trees and forests with mass/energy estimates. | wave packets, tree selection
82. Wave packet analysis | technique | Decomposes a solution/operator into space-frequency-localized wave packets traveling along light rays. | phase-space localization, uncertainty
83. Fefferman–Stein sharp maximal function theory | theorem | The sharp maximal function controls Lᵖ norms and vector-valued maximal inequalities hold, underpinning H¹–BMO duality. | maximal function, BMO
84. Stationary phase / steepest descent | technique | Estimates oscillatory integrals by their behavior at critical points of the phase, giving sharp decay rates. | Morse theory of phase
85. Sparse domination | technique | Bounds an operator pointwise by a positive sparse sum over a sparse family of cubes, yielding sharp weighted inequalities. | sparse families, Ap weights
86. Energy method (a priori energy estimates) | technique | Multiply a PDE by the solution or derivatives and integrate to derive controlled L²-type norms. | integration by parts, Grönwall
87. Moser iteration | technique | Iterating reverse-Hölder/Caccioppoli inequalities over shrinking balls to yield local boundedness and Harnack inequalities. | Sobolev embedding, Caccioppoli
88. Schauder estimates | theorem | Solutions of elliptic equations with Hölder coefficients gain two Hölder derivatives. | Hölder spaces, perturbation
89. Lᵖ elliptic (Calderón–Zygmund) estimates | theorem | Solutions of elliptic equations satisfy ‖D²u‖_{Lᵖ} ≲ ‖f‖_{Lᵖ} for 1<p<∞. | singular integrals
90. Strichartz estimates | theorem | Space-time Lᵖ_t L^q_x bounds for solutions of dispersive equations from L² initial data. | dispersive decay, TT*
91. Profile decomposition | technique | Writes a bounded sequence as a sum of asymptotically orthogonal rescaled/translated profiles plus a small remainder. | concentration-compactness, scaling
92. Bootstrap / continuity argument | technique | Assumes an estimate holds on a maximal interval then improves it strictly, forcing the interval to be the whole domain. | connectedness, openness/closedness
93. Compensated compactness (Murat–Tartar) | technique | Passes to limits in nonlinear quantities of weakly convergent sequences using div-curl structure and Young measures. | div-curl, Young measures
94. Div-curl lemma | theorem | If uₙ⇀u with bounded divergence and vₙ⇀v with bounded curl then uₙ·vₙ⇀u·v distributionally. | weak convergence, compact embedding
95. Aubin–Lions–Simon compactness lemma | theorem | Bounds on a function and its time derivative give strong time compactness for evolution PDE. | compact embedding, interpolation
96. Caffarelli–Silvestre extension | technique | Realizes the fractional Laplacian as a Dirichlet-to-Neumann map of a degenerate elliptic extension in one extra dimension. | degenerate elliptic, harmonic extension
97. Harnack inequality | theorem | Nonnegative solutions of elliptic/parabolic equations satisfy sup ≤ C inf on interior compacts. | Moser iteration, Krylov–Safonov
98. Krylov–Safonov estimates | theorem | Solutions of nondivergence-form uniformly elliptic equations with measurable coefficients are Hölder continuous with Harnack. | ABP maximum principle, covering
99. Vanishing viscosity method | technique | Constructs solutions of conservation laws / Hamilton–Jacobi equations as limits with a small added diffusion term. | entropy/BV bounds, compactness
100. Mountain pass theorem (Ambrosetti–Rabinowitz) | theorem | A C¹ functional with mountain-pass geometry and Palais–Smale has a critical point at the min-max level. | Palais–Smale, deformation lemma
101. Ekeland's variational principle | theorem | A lsc bounded-below functional admits points that are nearly minimizing and nearly critical. | completeness, lsc
102. Lyapunov–Schmidt reduction | technique | Splits a nonlinear equation into kernel/cokernel parts, reducing to a finite-dimensional problem. | Fredholm operators, IFT
103. Nehari manifold method | technique | Constrains a functional to the manifold where its derivative annihilates the solution direction, giving ground states. | fibering maps, constrained minimization
104. Γ-convergence (De Giorgi) | technique | A variational notion of functional convergence guaranteeing convergence of minimizers and minimum values. | lsc liminf inequality, recovery sequences
105. Direct method of the calculus of variations | technique | Proves existence of minimizers via coercivity plus weak lower semicontinuity on a reflexive space. | weak compactness, quasiconvexity
106. Ljusternik–Schnirelmann category method | technique | Uses topological invariants (category, genus) to count critical points of symmetric functionals. | Palais–Smale, equivariant topology
107. Pseudodifferential calculus | technique | Represents operators via symbols with a composition/adjoint calculus allowing parametrix construction modulo smoothing. | symbol classes, oscillatory integrals
108. Semiclassical analysis | technique | Studies the ℏ→0 limit linking quantum operators to classical Hamiltonian dynamics via ℏ-dependent symbol calculus. | semiclassical calculus, Weyl quantization
109. Propagation of singularities (Hörmander) | theorem | Singularities of solutions to pseudodifferential equations propagate along bicharacteristics of the principal symbol. | wavefront set, Hamilton flow
110. Stein's method | technique | Bounds distributional distance to a target law via a characterizing operator and the solution of an associated Stein equation. | Stein equation, exchangeable pairs
111. Malliavin calculus | technique | A differential calculus on Wiener space giving integration-by-parts formulas for densities and CLTs. | Wiener chaos, Skorokhod integral
112. Lindeberg exchange / replacement method | technique | Proves universality/CLTs by swapping summands one at a time and Taylor-controlling the change. | Taylor expansion, moment matching
113. Martingale CLT | theorem | Normalized martingale-difference sums with conditional variances converging to a constant converge to a Gaussian. | martingale differences, conditional Lindeberg
114. Concentration of measure | technique | Lipschitz/bounded-difference functions of many weakly dependent variables deviate from their mean with exponentially small probability. | bounded-difference, isoperimetry
115. Chaining / Dudley entropy bound | technique | Bounds the supremum of a stochastic process via successively finer approximations weighted by metric entropy. | covering numbers, sub-Gaussian increments
116. Second moment method | technique | Shows a nonnegative random variable is positive w.h.p. via P(X>0) ≥ (EX)²/E[X²]. | Paley–Zygmund, variance
117. Large deviations / Varadhan's lemma | theorem | Rare-event probabilities decay exponentially with a rate function; exponential integrals are governed by a variational principle. | rate function, Gärtner–Ellis
118. Schramm–Loewner evolution (SLE) | technique | Encodes conformally invariant random curves as solutions of the Loewner ODE driven by Brownian motion. | Loewner equation, conformal invariance
119. Gaussian free field | state | The Gaussian field with covariance the Green's function of the Laplacian, scaling limit of many lattice height models. | Laplacian Green's function, Gaussian measure
120. Regularity structures (Hairer) | technique | Local Taylor-like expansion and renormalization framework solving singular SPDEs driven by very rough noise. | models, reconstruction theorem, BPHZ
121. Paracontrolled calculus (Gubinelli–Imkeller–Perkowski) | technique | Solves singular SPDEs by expanding solutions as paraproducts controlled by explicit noise terms. | paraproducts, commutator estimates
122. KPZ universality / fluctuation theory | state | Universal t^{1/3} fluctuations and Tracy–Widom statistics of random interface growth in 1+1 dimensions. | KPZ equation, exact solvability
123. Stochastic quantization | technique | Constructs Euclidean QFT measures as invariant measures of a Langevin SPDE with added noise. | Langevin SPDE, renormalization
124. KAM theory | theorem | Most invariant tori of an integrable Hamiltonian persist under small Diophantine perturbation. | Newton iteration, small divisors
125. Hyperbolic / Anosov methods (shadowing) | technique | Uniform hyperbolicity yields stable/unstable foliations, structural stability and shadowing of pseudo-orbits. | hyperbolic splitting, cone fields
126. Thermodynamic formalism / transfer operator | technique | Studies invariant measures and decay of correlations via spectral properties of the Ruelle transfer operator. | transfer operator, pressure
127. Pesin theory (nonuniform hyperbolicity) | theorem | Positive Lyapunov exponents yield stable/unstable manifolds and an entropy formula without uniform hyperbolicity. | Lyapunov exponents, Oseledets
128. Oseledets multiplicative ergodic theorem | theorem | For an ergodic cocycle, Lyapunov exponents and an invariant filtration/splitting exist almost everywhere. | subadditive ergodic theorem, cocycle
129. Ergodic decomposition theorem | theorem | Any invariant measure is a unique integral of ergodic invariant measures. | Choquet theory, conditional measures
130. Ratner's theorems (unipotent flows) | theorem | Orbit closures and invariant measures of unipotent flows on homogeneous spaces are algebraic. | unipotent dynamics, measure rigidity
131. Renormalization in dynamics | technique | Studies universality/rigidity by iterating a renormalization operator with a hyperbolic fixed point. | renormalization operator, self-similarity
132. Cluster expansion (Mayer expansion) | technique | Expands the log-partition function as a convergent sum over connected clusters at high temperature/low density. | polymer graphs, Kotecký–Preiss
133. Reflection positivity | technique | A positivity under reflection yielding chessboard estimates, infrared bounds and proofs of phase transitions. | Osterwalder–Schrader, Gaussian domination
134. Lee–Yang theorem | theorem | Zeros of the partition function as a function of complex field lie on the imaginary axis / unit circle. | partition function, analyticity
135. Replica / cavity method | technique | Computes free energies of disordered spin-glass systems via the replica limit or self-consistent cavity recursions. | replica symmetry breaking, interpolation
136. Log-Sobolev inequality / hypercontractivity | theorem | A log-Sobolev inequality is equivalent to hypercontractivity of the semigroup and gives exponential ergodicity. | Dirichlet form, entropy-energy
137. Infrared bound / continuous symmetry breaking | technique | Bounds the Fourier-mode two-point function to prove long-range order and spontaneous breaking of continuous symmetry in d≥3. | reflection positivity, spin-wave
138. Constructive QFT / phase-cell expansion (Glimm–Jaffe) | technique | Builds interacting Euclidean field measures by multiscale phase-cell/cluster expansions with renormalization. | cluster expansion, RG
139. Renormalization group (Wilson–Kadanoff) | technique | Iteratively integrates out short-distance degrees of freedom and rescales, flowing the effective action toward fixed points. | block-spin coarse-graining, fixed points
140. Bakry–Émery Γ-calculus | technique | Derives functional inequalities and convergence rates from a curvature-dimension lower bound CD(K,N). | carré du champ, diffusion semigroup
141. Mountain-pass linking / index theory | technique | Min-max critical point theory using linking geometry and topological index for indefinite functionals. | Palais–Smale, linking sets

## CLUSTER C — Combinatorics / Geometry / Topology / Logic / TCS / Optimization

142. Shearer's entropy lemma | theorem | Entropy of a joint distribution is bounded by averaged entropies of a covering family of marginals. | subadditivity of entropy
143. Graph removal lemma | theorem | A graph with few copies of a fixed subgraph H can be made H-free by deleting few edges. | regularity lemma, counting lemma
144. Hypergraph regularity method | technique | Higher-order regularity partitions plus counting lemma extending Szemerédi regularity to k-uniform hypergraphs. | regularity lemma, hypergraph counting
145. Rödl nibble (semi-random method) | technique | Build a structure in small random increments tracking quantities by concentration to achieve near-optimal packings. | probabilistic method, concentration
146. Kruskal–Katona theorem | theorem | Bounds the number of (k−1)-shadows of a family of k-sets, characterizing extremal set systems by colex order. | shadows, compression
147. Dependent random choice | technique | Find a large vertex subset whose every small tuple has many common neighbors via random common-neighbor sampling. | probabilistic method, convexity
148. Stepping-up lemma | technique | Construct colorings on larger ground sets from lower-bound colorings to push Ramsey lower bounds up exponentially. | Ramsey theory, partition calculus
149. Algebraic shifting / Stanley–Reisner method | technique | Associate to a simplicial complex a squarefree monomial ideal and shifted complex to study f-vectors via commutative algebra. | commutative algebra, simplicial complexes
150. Topological combinatorics / Borsuk–Ulam method | technique | Reduce combinatorial existence statements to topological obstructions via Borsuk–Ulam and equivariant maps. | Borsuk–Ulam, equivariant topology
151. Discharging method | technique | Assign initial charges via Euler's formula then redistribute to force a forbidden configuration in planar graphs. | Euler's formula, planar graphs
152. h-principle (Gromov) | technique | Reduce a partial differential relation to its underlying homotopy-theoretic problem, solving geometric PDEs by homotopy. | jet bundles, convex integration
153. Surgery theory | technique | Modify a manifold by cutting out and regluing handle neighborhoods to control topology, via the surgery exact sequence. | cobordism, L-theory
154. Cobordism theory | technique | Classify manifolds up to bounding a higher-dimensional manifold, computed via Thom spectra. | Thom transversality, characteristic classes
155. Ricci flow (Hamilton–Perelman) | technique | Evolve a Riemannian metric by its Ricci curvature with surgery to control singularities and homogenize geometry. | parabolic PDE, entropy monotonicity
156. Almgren–Pitts min-max theory | technique | Produce minimal surfaces as saddle critical points of the area functional via min-max over sweepouts. | geometric measure theory, varifolds
157. Mean curvature flow | technique | Evolve a submanifold by its mean curvature vector to study singularities and topology. | geometric parabolic PDE, monotonicity formula
158. Gromov–Hausdorff convergence and collapsing | technique | Take limits of metric spaces in the Gromov–Hausdorff metric controlling degeneration via bounded geometry. | metric geometry, comparison geometry
159. Cheeger–Gromoll splitting / soul theorem | theorem | A nonnegatively curved manifold containing a line splits isometrically; noncompact ones retract to a soul. | comparison geometry, Busemann functions
160. Pseudoholomorphic curves (Gromov) | technique | Study symplectic manifolds via J-holomorphic maps from Riemann surfaces whose moduli yield invariants. | almost complex structures, compactness
161. Floer homology | technique | An infinite-dimensional Morse homology on an action/Chern–Simons functional whose critical points are geometric solutions. | pseudoholomorphic curves, gluing
162. Seiberg–Witten theory | technique | Smooth 4-manifold invariants from counting solutions of the Seiberg–Witten monopole equations on a spin-c structure. | gauge theory, Dirac operators
163. Heegaard Floer homology | technique | Invariants of 3-manifolds/knots from Lagrangian Floer homology of tori in a symmetric product of a Heegaard surface. | Heegaard diagrams, pseudoholomorphic curves
164. Donaldson gauge theory | technique | Study 4-manifolds via moduli spaces of anti-self-dual connections on principal bundles. | connections, Yang–Mills
165. Persistent homology | technique | Track birth/death of homology classes across a filtration to produce stable multiscale barcodes. | filtered homology, stability theorem
166. Normal surface theory | technique | Represent surfaces in a triangulated 3-manifold by intersections with tetrahedra as integer solutions to matching equations. | triangulations, integer programming
167. Iterated forcing | technique | Compose forcing extensions transfinitely with finite/countable support to satisfy many requirements simultaneously. | forcing, properness
168. Back-and-forth method | technique | Build an isomorphism/elementary map by alternately extending finite partial maps on both sides. | first-order logic, saturation
169. Stability theory / forking | technique | Classify theories and control independence of types via forking, a notion of generic dependence. | type spaces, model theory
170. Morley rank | technique | Assign an ordinal dimension to definable sets via Cantor–Bendixson analysis of type spaces, governing ω-stable theories. | stability, type spaces
171. Inner model theory / fine structure | technique | Construct canonical minimal models (L, extender models) analyzed level-by-level via fine structure. | constructibility, extenders
172. Priority argument / finite injury | technique | Construct c.e. sets meeting infinitely many requirements by prioritizing and allowing finitely many injuries each. | computability, c.e. sets
173. Idempotent ultrafilter method | technique | Use idempotent ultrafilters on semigroups to prove partition-regularity results such as Hindman's theorem. | ultrafilters, semigroup compactification
174. Lovász theta function / SDP relaxation | technique | Optimize a linear objective over PSD matrices; the theta function sandwiches clique and chromatic numbers. | semidefinite programming, duality
175. PCP theorem | theorem | Every NP statement has a proof checkable by reading O(1) random bits, implying hardness of approximation. | error-correcting codes, verifier composition
176. Spectral sparsification | technique | Approximate a graph's Laplacian by a sparse reweighted subgraph preserving all quadratic forms within (1±ε). | effective resistance, matrix concentration
177. Multiplicative weights update method | technique | Iteratively adjust weights of experts/constraints multiplicatively to converge to near-optimal decisions with regret bounds. | online learning, convexity
178. Sum-of-squares hierarchy (Lasserre) | technique | Approximate polynomial optimization by a hierarchy of SDP relaxations certifying nonnegativity as sums of squares. | SDP, Positivstellensatz
179. Smoothed analysis | technique | Analyze algorithm performance on slightly randomly perturbed worst-case inputs to explain practical efficiency. | probability, perturbation analysis
180. Atiyah–Singer index theorem | theorem | The analytic index of an elliptic operator on a closed manifold equals a topological index in characteristic classes. | elliptic operators, K-theory
181. Dependent / interlacing real-rootedness (MSS) | technique | Prove existence of a structure with bounded eigenvalues by finding a common interlacer of characteristic polynomials. | real-rooted polynomials, mixed char polynomials
