Glossary¶

Compact research-facing terms. School synonyms omitted when the advanced name is standard.

Term	Domain	Definition
Abelian category	HA	Additive cat. with kernels/cokernels; $\operatorname{coim}\xrightarrow{\sim}\operatorname{im}$
Adjunction	Cat	Pair $F\dashv G$ with unit/counit (or Hom-iso)
Algebraically closed	AA	Field where every nonconst poly has a root (e.g. $\mathbb{C}$)
Almost complex structure	DG	$J:TM\to TM$, $J^2=-I$
Atiyah–Singer index	DG/Anal	$\operatorname{ind} D=\langle\operatorname{ch}(\sigma_D)\operatorname{Td},[TM]\rangle$ (schematic)
Banach space	FA	Complete normed vector space
Basis	LA	Linearly independent spanning set
Birational map	AG	Iso after restricting to dense opens of varieties/schemes
Brownian motion	Prob	Cont. process, indep. Gaussian incr., $\langle B\rangle_t=t$
Capacity (channel)	IT	$C=\max_{p_X}I(X;Y)$ asymptotic coding rate limit
Cayley–Hamilton	LA	$\chi_A(A)=0$
Characteristic class	Top/DG	Cohomology classes obstructing structure (Chern, Pontryagin, …)
Compact	Top	Every open cover has a finite subcover
Complete	Anal/Top	Cauchy nets/sequences converge (metric: seq.)
Condition number	Num	Sensitivity of problem output to input perturbations
Connected	Top	Not union of two disjoint nonempty opens
Cup product	AlgTop	Graded product $H^p\times H^q\to H^{p+q}$ on cohomology
Derived category	HA	Localization of complexes at quasi-isomorphisms
Determinant	LA	Alternating multilinear volume $\det:\operatorname{End}\to K$
Discrete log	Crypto/NT	Invert $x\mapsto g^x$ in a group
Distribution (gen. fn)	Anal	Continuous linear functional on test functions
Double complex	HA	Bigraded object with commuting (anti)differentials $d_h,d_v$
Eigenvalue	LA	$\lambda$ with $Tv=\lambda v$, $v\neq0$
Entropy (Shannon)	IT	$H(X)=-\sum p\log p$; operational compression rate
Étale morphism	AG	Flat + unramified (alg. analogue of local iso)
Exact sequence	AA/Top	$\operatorname{im}=\ker$ at each spot
Ext	HA	$R^i\operatorname{Hom}$; classifies extensions for $i=1$
Field of fractions	AA	$\operatorname{Frac}(R)$ for domain $R$
Fredholm operator	FA	$\dim\ker,\dim\operatorname{coker}$ finite; index $\dim\ker-\dim\operatorname{coker}$
Fréchet derivative	Anal	Best continuous linear approximation
Galois group	AA	$\mathrm{Aut}(L/K)$ for Galois $L/K$
Generator (diffusion)	Prob/SDE	$\mathcal{L}=b\cdot\nabla+\frac12 a^{ij}\partial_i\partial_j$ of a Markov process
Group	AA	Set with assoc. op, identity, inverses
Haar measure	Anal	Translation-invariant measure on locally compact group
Hartman–Grobman	Dyn	Hyperbolic eq. topologically conjugate to linearization
Hilbert space	FA	Complete inner-product space
Homeomorphism	Top	Bicontinuous bijection
Hyperbolic equilibrium	Dyn	$Df(x_*)$ has no eigenvalue with $\operatorname{Re}\lambda=0$
Ideal	AA	Additive subgroup absorbing ring mult.
IND-CPA	Crypto	Ciphertext indistinguishability under chosen plaintexts
Itô integral	Prob	Stochastic integral $\int H\,dB$ for progressive $H$
Jacobian	Calc	Matrix / det of $Df$
Jordan form	LA	Canonical upper-triangular block form over split $\chi$
Kähler manifold	DG	Complex manifold with closed positive $(1,1)$ form
KKT conditions	Opt	Stationarity + primal/dual feasibility + complementary slackness
KL divergence	IT/Prob	$D_{\mathrm{KL}}(p\\|q)=\sum p\log(p/q)$
Langlands correspondence	NT	Conjectural $\pi\leftrightarrow\rho$ matching $L$-functions
Lebesgue integral	Measure	Integration w.r.t. complete translation-invariant measure on $\mathbb{R}^n$
Lyapunov function	Dyn	Energy-like $V$ with $\dot V\le0$ controlling stability
Manifold	DG/Top	Locally Euclidean Hausdorff second-countable space
Martingale	Prob	$\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$ adapted integrable process
Measure	Measure	Countably additive $[0,\infty]$-valued set function on a $\sigma$-algebra
Module	AA	Abelian group with ring action (vector space if ring is a field)
Nash equilibrium	GT	Strategy profile with no profitable unilateral deviation
Noetherian	AA	Ascending chain condition on ideals
Norm	FA	$\\|x\\|\ge0$, definite, homogeneous, triangle
Normal subgroup	AA	$gHg^{-1}=H$; kernel of some hom
$p$-adic valuation	NT	$\\|a\\|_p=p^{-v_p(a)}$ nonarchimedean absolute value
Plancherel theorem	FA/HA	Fourier transform unitary on $L^2$ (fixed normalization)
Poincaré duality	AlgTop	$H^k(M)\cong H_{n-k}(M)$ for closed oriented $n$-manifolds
Radon–Nikodym derivative	Measure	$d\nu/d\mu$ when $\nu\ll\mu$
Rank–nullity	LA	$\operatorname{rank}T+\operatorname{nullity}T=\dim V$
Residue	Complex	Coeff of $(z-a)^{-1}$ in Laurent series
Riesz representation	FA/Measure	Dual of $C(K)$ / Hilbert $H^*$ via measures / inner product
Ring	AA	$(R,+)$ abelian group + mult. + distributivity
Scheme	AG	Locally ringed space locally $\operatorname{Spec} R$
Spectral radius	FA	$r(T)=\sup_{\lambda\in\sigma(T)}
Spectral sequence	HA	Pages $E_r$ with $d_r$ abutting to graded pieces of a target
Spectrum (operator)	FA	$\{\lambda:T-\lambda I\text{ not invertible}\}$
Spectrum (ring)	AA	$\operatorname{Spec} R=$ prime ideals
Stack (alg.)	AG	Fibered category with descent (moduli of objects with autos)
Stokes theorem	DG	$\int_M d\omega=\int_{\partial M}\omega$
Subdifferential	Opt	$\partial f(x)=\{v:f(y)\ge f(x)+\langle v,y-x\rangle\ \forall y\}$
Tensor product	AA/LA	Universal bilinear target $V\otimes W$
Topology	Top	Family of opens closed under $\cup$ and finite $\cap$
Tor	HA	$L_i(-\otimes N)$; measures failure of flatness / torsion
UFD / PID / Euclidean	AA	Unique factorization / principal ideals / division algorithm domains
Uniform convergence	Anal	$\\|f_n-f\\|_\infty\to0$; preserves continuity
Valuation	NT/AA	Hom $K^\times\to\Gamma$ with ultrametric inequality
Variety (affine)	AG	Zero set of polynomials in $K^n$ (classical)
Vector space	LA	Module over a field
Weak topology	FA	Coarsest topology making dual functionals continuous
Whitehead theorem	AlgTop	Weak homotopy eq. of CW complexes $\Rightarrow$ homotopy eq.
Yoneda lemma	Cat	$\operatorname{Nat}(h_X,F)\cong F(X)$; embed $C\hookrightarrow\operatorname{PSh}(C)$
$\sigma$-algebra	Measure	Closed under complement and countable union
Almost sure	Prob	Holds outside a set of probability zero
Artin $L$-function	NT	$L(s,\rho)$ from continuous Galois rep $\rho$ via charpoly of Frobenius
Chow group	AG	Cycles modulo rational equivalence; $A_*(X)$
Cobordism	AlgTop	Manifolds modulo boundary relation; spectrum $MO$, $MU$, …
Crystalline cohomology	AG/NT	$p$-adic cohomology for schemes over perfect fields of char $p$
Deformation quantization	MathPhys	Star product $f\star g=fg+\hbar\{f,g\}/2+O(\hbar^2)$ on Poisson manifold
Deligne–Mumford stack	AG	Stack with unramified diagonal + proper schematic atlas (moduli of curves)
Floquet multiplier	DE	Eigenvalue of monodromy matrix for periodic linear ODE
Free independence	Prob	Vanishing of mixed free cumulants; noncommutative independence
Geometric Langlands	NT/AG	Hecke eigensheaves on $\mathrm{Bun}_G$ $\leftrightarrow$ $G^\vee$-local systems
Gromov–Witten invariant	AG/SG	Counts of stable maps $C\to X$ via virtual classes on moduli
Harder–Narasimhan filtration	AG	Canonical filtration of vector bundle by decreasing slopes
Hörmander condition	Anal/SDE	Lie algebra of diffusion fields spans $\Rightarrow$ hypoellipticity / smooth densities
Instanton	Gauge	Self-dual YM connection; absolute minimizer of YM energy on $S^4$
Model category	Cat/Homotopy	Category with weak equivalences, fibrations, cofibrations satisfying Quillen axioms
Motivic cohomology	AG	Bigraded cohomology of schemes (Voevodsky); related to $K$-theory via AT
Operad	Cat	Algebraic structure encoding composable $n$-ary operations ($\mathbb{E}_n$, $\mathrm{Ass}$, …)
Perverse sheaf	AG/Top	Intermediate-extension $t$-structure heart on stratified spaces; IC complexes
Picard group	AG/AA	$\operatorname{Pic}(X)=$ line bundles up to iso $\cong H^1(X,\mathcal{O}_X^\times)$
Quillen adjunction	Cat	Adjunction where left preserves cofibrations and acyclic cofibrations
Rough path	Prob	Path plus iterated integrals satisfying Chen / analytic bounds ($p$-variation)
Satake isomorphism	Rep/NT	$H_K\bigl(G(F)/\!/K\bigr)\cong\mathbb{C}[\widehat{G}]^{\widehat{G}}$ (unramified Hecke)
Semicircle law	Prob/RMT	Wigner limiting spectral density $\frac{1}{2\pi}\sqrt{4-x^2}$
Symplectic form	DG	Closed nondegenerate $2$-form $\omega$; Hamiltonian mechanics geometry
Thick subcategory	HA/Cat	Triangulated subcategory closed under direct summands
Tomita–Takesaki	FA/Phys	Modular automorphism group of a von Neumann algebra with cyclic separating vector
Tracy–Widom law	Prob/RMT	Limiting distribution of rescaled largest eigenvalue at soft edge
Verdier duality	AG/Top	Duality $f^!\leftrightarrow f_!$ / dualizing complex on locally compact spaces
Virtual fundamental class	AG	Replacement for $[M]$ when moduli stack is not smooth of expected dim
Witt vectors	AA/NT	Ring scheme $W(k)$ lifting char-$p$ perfect $k$; $p$-typical $W(k)$