Mathematics Cheat Sheet¶

Recall (doc/postdoc). Formula density only — hypotheses tacit unless stated. Topic pages for obstructions.

Arithmetic / fields¶

Formula	Name
$a=bq+r$, $0\le r<b$	Division algorithm in $\mathbb{Z}$
$a(b+c)=ab+ac$	Distributivity
$a^ma^n=a^{m+n}$, $(a^m)^n=a^{mn}$	Exponent monoid laws
$a^{-1}$ iff unit; $\operatorname{Frac}(R)$	Inverses / localization
$\log_b(xy)=\log_b x+\log_b y$	Log group iso
$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$	Standard tower

Algebra¶

Formula	Name
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$	Quadratic formula (char $\neq2$)
$\Delta=b^2-4ac$	Discriminant
$a^2-b^2=(a-b)(a+b)$	Difference of squares
$y-y_1=m(x-x_1)$	Point-slope (affine line)
$f:A\to B$ inj/surj/bij	Set-map classes
$\log_b x=y\iff b^y=x$	Log definition
$K[x]$ Euclidean if $K$ field	Poly division

Geometry / trig¶

Formula	Name
$c^2=a^2+b^2-2ab\cos C$	Law of cosines
$\frac{a}{\sin A}=2R$	Law of sines
$A+B+C=\pi$	Euclidean triangle
$\sin^2\theta+\cos^2\theta=1$	Circle identity
$e^{i\theta}=\cos\theta+i\sin\theta$	Euler
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$	Addition
$(x-h)^2+(y-k)^2=r^2$	Circle
$V=\frac43\pi r^3$, $A=4\pi r^2$	Sphere

Calculus¶

Formula	Name
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$	Derivative
$(fg)'=f'g+fg'$, $(f\circ g)'=(f'\circ g)g'$	Product / chain
$\int_a^b f=F(b)-F(a)$	FTC
$\int u\,dv=uv-\int v\,du$	Parts
MVT: $f'(c)=\frac{f(b)-f(a)}{b-a}$	Mean value
Taylor + Lagrange remainder	Expansion
$\sum n^{-p}<\infty\iff p>1$	$p$-series
$R^{-1}=\limsup\\|c_n\\|^{1/n}$	Power series radius
$\nabla\times\nabla f=0$, $\nabla\cdot(\nabla\times F)=0$	Vector calc
$\int_{\partial M}\omega=\int_M d\omega$	Stokes (forms)

Linear algebra¶

Formula	Name
$T\in L(V,W)$; $[T]_{B'}=P^{-1}[T]_BP$	Linear maps / change of basis
$\operatorname{rank}T+\operatorname{nullity}T=\dim V$	Rank–nullity
$\det(AB)=\det A\det B$	Multiplicativity
$A^{-1}=\frac1{\det A}\operatorname{adj}A$	Adjugate inverse
$\chi_A(t)=\det(tI-A)$	Characteristic poly
$Av=\lambda v$; $\chi_A(A)=0$	Eigen / Cayley–Hamilton
$A=PDP^{-1}$ iff diagonalizable	Spectral form
$\langle u,v\rangle$; $\\|v\\|=\sqrt{\langle v,v\rangle}$	Inner product
$\\|u\times v\\|=\\|u\\|\\|v\\|\sin\theta$	Cross (3D)

Abstract algebra¶

Formula	Name
Group: assoc., $e$, $a^{-1}$	Group axioms
$\\|H\\|\mid\\|G\\|$ (finite)	Lagrange
$G/\ker\phi\cong\operatorname{im}\phi$	1st iso. thm
Ring / field / $R$-mod	Algebra hierarchy
$K(x)=\operatorname{Frac}(K[x])$	Rational functions
$\mathrm{Gal}(L/K)$ acts on roots	Galois action

Topology / analysis¶

Formula	Name
$\emptyset,X\in\tau$; arbitrary $\cup$; finite $\cap$	Topology
$f$ cont. $\iff f^{-1}(\text{open})$ open	Continuity
Compact: every open cover has finite subcover	Heine–Borel on $\mathbb{R}^n$
$d(x,z)\le d(x,y)+d(y,z)$	Metric
Complete: Cauchy $\Rightarrow$ convergent	Banach setup
$V-E+F=2$	Euler ($S^2$)

DE / PDE¶

Formula	Name
$y'+Py=Q$: $\mu=e^{\int P}$	1st-order linear
$r^2+br+c=0$	Const-coeff characteristic
$\mathcal{L}\{y'\}=sY-y(0)$	Laplace
$u_t=\alpha u_{xx}$, $u_{tt}=c^2u_{xx}$, $\Delta u=0$	Heat / wave / Laplace
$\Delta=B^2-4AC$ type	2nd-order PDE class

Complex analysis¶

Formula	Name
CR: $u_x=v_y$, $u_y=-v_x$	Holomorphy (with $C^1$)
Cauchy: $\int_{\partial\Omega}f=0$ (holomorphic)	Cauchy theorem
$f(a)=\frac1{2\pi i}\oint\frac{f(z)}{z-a}dz$	Cauchy integral
$\oint f=2\pi i\sum\operatorname{Res}$	Residue theorem
$e^z$, $\operatorname{Log}$ branched	Entire / multi-valued

Probability / statistics¶

Formula	Name
$\mathbb{P}(A\mid B)=\mathbb{P}(A\cap B)/\mathbb{P}(B)$	Conditional
Bayes: $\mathbb{P}(A\mid B)\propto\mathbb{P}(B\mid A)\mathbb{P}(A)$	Bayes
$\mathbb{E}[X]=\int x\,dF$; $\operatorname{Var}=\mathbb{E}X^2-(\mathbb{E}X)^2$	Moments
$X\sim N(\mu,\sigma^2)$: density $\frac1{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$	Gaussian
CLT / LLN	Asymptotics (schematic)

Number theory / crypto¶

Formula	Name
$a\equiv b\pmod m\iff m\mid(a-b)$	Congruence
$\gcd(a,b)=\gcd(b,a\bmod b)$	Euclid
$a^{p-1}\equiv1\pmod p$ ($p\nmid a$)	Fermat
$a^{\varphi(n)}\equiv1\pmod n$ ($\gcd(a,n)=1$)	Euler
RSA: $ed\equiv1\pmod{\varphi(n)}$	RSA exponent
DLP: $g^x=h$ in group	Discrete log

Functional analysis / operator spectrum¶

Formula	Name
Norm axioms; Banach = complete	Normed spaces
Hilbert: complete inner-product	Hilbert
$\\|Tx\\|\le M\\|x\\|$	Bounded operator
Riesz: $H^*\cong H$ (Hilbert)	Dual (sketch)
$\sigma(T)=\{\lambda:T-\lambda\notin\mathcal{L}(X)^{-1}\}$	Spectrum
$\sigma(T)\neq\emptyset$ compact (complex Banach alg.)	Gelfand nonempty
$r(T)=\sup	\lambda
$T=T^*$ on Hilbert: $\sigma(T)\subset\mathbb{R}$	Self-adjoint spectrum
Compact $K$: $\sigma(K)\subseteq\{0\}\cup$ evals $\to0$	Compact spectrum
$I-K$ Fredholm: inj $\Leftrightarrow$ surj	Fredholm alternative
Spectral thm: normal $T$ $\cong$ mult. by $\lambda$ on $L^2(\mu)$	Multiplication form

Algebraic geometry / schemes¶

Formula	Name
$\operatorname{Spec} R=\{\mathfrak{p}\text{ prime}\}$ + Zariski	Prime spectrum
$\mathcal{O}_{\operatorname{Spec} R}(D(f))=R_f$	Structure sheaf
Scheme: loc. ringed space loc. $\operatorname{Spec} R$	Scheme
$X\times_S Y$ fiber product	Base change
$\operatorname{Mor}(X,\operatorname{Spec} A)\cong\operatorname{Hom}(A,\Gamma(X,\mathcal{O}_X))$	Affine morphisms
$H^i(X,\mathcal{F})=R^i\Gamma(X,\mathcal{F})$	Sheaf cohomology
Serre: $H^i(X,\mathcal{F}(n))=0$ ($i>0$, $n\gg0$) on $\mathbb{P}^n$	Serre vanishing
$\omega_X$ dualizing / canonical sheaf	Serre duality setup
Smooth / étale / flat morphisms	Relative regularity
$\dim \mathcal{O}_{X,x}=$ Krull dim of local ring	Dimension

Homological algebra — Ext / Tor¶

Formula	Name
$L_iF(M)=H_i(F(P_\bullet))$	Left derived
$R^iG(M)=H^i(G(I^\bullet))$	Right derived
$\operatorname{Tor}_i^R(M,N)=L_i(M\otimes_R N)$	Tor
$\operatorname{Ext}^i_R(M,N)=R^i\operatorname{Hom}_R(M,N)$	Ext
$\operatorname{Ext}^1(C,A)\cong$ ext. classes $0\to A\to B\to C\to0$	Yoneda $\operatorname{Ext}^1$
$\operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/n)\cong\mathbb{Z}/\gcd$	Tor of cyclics
$\operatorname{pd} M=\max\{i:\operatorname{Ext}^i(M,-)\neq0\}$	Projective dim
$\operatorname{gl.dim} R=\sup_M\operatorname{pd} M$	Global dimension
Hilbert syzygy: $\operatorname{gl.dim} k[x_1,\ldots,x_n]=n$	Syzygy theorem
$H^i(G,M)=\operatorname{Ext}^i_{\mathbb{Z}G}(\mathbb{Z},M)$	Group cohomology

Spectral sequences¶

Formula	Name
$(E_r,d_r)$, $E_{r+1}=H(E_r)$	SS pages
$d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$	Cohomological $d_r$
$E_r\Rightarrow H^{p+q}$ (filtered)	Convergence
$R^pF(R^qG)\Rightarrow R^{p+q}(FG)$	Grothendieck SS
$H^p(G/N;H^q(N;M))\Rightarrow H^{p+q}(G;M)$	LHS / Lyndon
$H^p(B;H^q(F))\Rightarrow H^{p+q}(E)$	Serre SS (fib.)
$\operatorname{Tot}(C^{\bullet,\bullet})$, two filtrations	Double complex $\Rightarrow$ SS
First quadrant: finite $d_r$ hits	Edge maps exist

Stochastic processes / SDE¶

Formula	Name
$(\mathcal{F}_t)$ filtration; $\tau$ stopping time	Information flow
$M_t$ martingale: $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$	Martingale
BM: cont. paths, indep. Gauss incr., $\langle B\rangle_t=t$	Brownian motion
$\int H\,dB$ Itô isometry $\mathbb{E}(\int H\,dB)^2=\mathbb{E}\int H^2$	Itô integral
$dX=b\,dt+\sigma\,dB$	SDE (Itô)
Itô: $df=f_t dt+f_x dX+\frac12 f_{xx}d\langle X\rangle$	Itô formula
Lipschitz $b,\sigma\Rightarrow$ strong existence/uniqueness	SDE Picard
Generator $\mathcal{L}=b\cdot\nabla+\frac12\operatorname{tr}(\sigma\sigma^\top D^2)$	Diffusion generator
Feynman–Kac: $u=\mathbb{E}[\cdots]$ solves parabolic PDE	Prob. representation
Girsanov: change drift $\Leftrightarrow$ reweight by exp mart.	Measure change

Langlands (placeholder one-liners)¶

Formula	Name
$L(\pi,s)$ automorphic $L$-function	Automorphic $L$
$L(\rho,s)$ from Galois rep $\rho:\mathrm{Gal}_K\to\mathrm{GL}_n$	Galois $L$
Global Langlands: $\pi\leftrightarrow\rho$ matching $L$-factors	Reciprocity (conj.)
Local Langlands: $\operatorname{Irr}(G(K))\leftrightarrow$ Weil–Deligne reps	LLC
Modularity: elliptic curves $/\mathbb{Q}\leftrightarrow$ wt 2 cusp forms	Wiles et al.
Functoriality: $L$-homomorphisms induce $\pi\mapsto\pi'$	Langlands functoriality
Geometric Langlands: $\operatorname{Bun}_G$ Hecke eigensheaves $\leftrightarrow$ loc. sys.	Geometric form
Trace formula $\leftrightarrow$ spectral side of automorphic forms	Arthur–Selberg

Symplectic geometry¶

Formula	Name
$\omega\in\Omega^2(M)$ closed, nondeg.	Symplectic form
$\iota_{X_H}\omega=-dH$ (or $+$; sign conv.)	Hamiltonian vector field
$\{f,g\}=\omega(X_f,X_g)$	Poisson bracket
$\mathcal{L}_{X_H}\omega=0$	Hamiltonian flow preserves $\omega$
$dH(X_H)=0$	Energy conservation
$\omega_0=\sum dq_i\wedge dp_i$ on $T^*Q$	Canonical / Liouville form setup
$\mu:M\to\mathfrak{g}^*$, $\langle d\mu_\xi,v\rangle=\omega(X_\xi,v)$	Moment map
$\mu^{-1}(0)/G$ (reg., free) symplectic	Marsden–Weinstein reduction
$\omega^n/n!$	Liouville volume
Lagrangian $L$: $\omega	_L=0$, $\dim L=\frac12\dim M$
$f^*\omega=\omega$	Symplectomorphism
Darboux: locally $\omega=\omega_0$	Canonical coordinates

Random matrices¶

Formula	Name
GOE/GUE/GSE: dens. $\propto e^{-\frac{\beta}{4}\operatorname{tr}H^2}$	Gaussian ensembles ($\beta=1,2,4$)
$P(\lambda)\propto\prod_i e^{-\beta\lambda_i^2/4}\prod_{i<j}\\|\lambda_i-\lambda_j\\|^\beta$	Joint eigenvalue law
$\rho_{\mathrm{sc}}(x)=\frac{1}{2\pi}\sqrt{4-x^2}$ on $[-2,2]$	Wigner semicircle (std. scaling)
$K_N(x,y)$ sine/Airy/… kernels	Correlation kernels
$\det\bigl(K(x_i,x_j)\bigr)$	Gap / $n$-point probs (determinantal)
$\frac{\lambda_{\max}-2}{N^{-2/3}}\Rightarrow\mathrm{TW}_\beta$	Tracy–Widom edge
$N^{-1}\sum f(\lambda_i)\to\int f\rho$	Empirical spectral measure
Haar unitary: $U(N)$ with Haar measure	CUE prototype
$\mathbb{E}	p_U(z)
Marchenko–Pastur: $\rho(x)=\frac{\sqrt{(b-x)(x-a)}}{2\pi\sigma^2 x}$	Wishart / covariance spectra
Free convolution $\boxplus$ of spectral laws	Additive free convolution

Free probability¶

Formula	Name
$(\mathcal{A},\varphi)$ noncomm.\ prob.\ space	$\varphi$ tracial state (often)
$a_1,\ldots,a_n$ free if mixed free cumulants vanish	Freeness
$R_a(z)=\sum\kappa_n(a)z^{n-1}$	$R$-transform (free cumulants)
$R_{a+b}=R_a+R_b$ when $a,b$ free	Free additivity
$G_a(z)=\varphi((z-a)^{-1})$	Cauchy / Stieltjes transform
$G_a^{\langle-1\rangle}(w)=1/w+R_a(w)$	$R$ via inverse of $G$
Semicircle: $R(z)=z$ (std.); $G=(z-\sqrt{z^2-4})/2$	Free CLT law
Free Poisson / Marchenko–Pastur	Free compound Poisson
$S$-transform: mult. free convolution $\boxtimes$	Multiplicative freeness
$\kappa_n$ free cumulants via Möbius on N-C partitions	Speicher combinatorial formula
Asymptotic freeness of independent Wigner/Haar	Voiculescu (RMT link)
$\varphi(a^*a)\ge0$; $\\|\cdot\\|_\varphi$	$L^2(\varphi)$ GNS setup

Model categories / homotopy algebra¶

Formula	Name
$(\mathbf{C},\mathcal{W},\mathcal{C},\mathcal{F})$ model cat.	Weak eq. / cofibrations / fibrations
2-of-3 for $\mathcal{W}$; lift axioms; factorization	Quillen model axioms (sketch)
$X\xrightarrow{\sim}RX$ fibrant repl.; $QX\xrightarrow{\sim}X$ cofibr.	Replacements
$\mathbf{Ho}(\mathbf{C})=\mathbf{C}[\mathcal{W}^{-1}]$	Homotopy category
$\operatorname{Map}(QX,RY)$	Derived mapping space (sketch)
$F$ left Quillen: preserves cofibrations + acyclic cofibr.	Quillen functor
$LF\dashv RG$ on homotopy cats	Derived adjunction
Cofibrant $X$: $\operatorname{Hom}(X,-)$ preserves weak eq. betw. fibrants	Homotopy-correct Hom
$\mathbf{sSet}$, $\mathbf{Top}$, $\mathrm{Ch}_{\ge0}(R)$	Standard model structures
$f$ weak eq. of fibrant-cofibrant $\Rightarrow$ homotopy eq.	Whitehead-type
Cylinder $X\sqcup X\to X\wedge I\to X$; path $Y\to Y^I$	Left/right homotopies
$\infty$-cat localization $N(\mathbf{C})[\mathcal{W}^{-1}]$	Higher upgrade (pointer)

Formula	Name
\((E_r,d_r)\), \(E_{r+1}=H(E_r)\)	SS pages
\(d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}\)	Cohomological \(d_r\)
\(E_r\Rightarrow H^{p+q}\) (filtered)	Convergence
\(R^pF(R^qG)\Rightarrow R^{p+q}(FG)\)	Grothendieck SS
\(H^p(G/N;H^q(N;M))\Rightarrow H^{p+q}(G;M)\)	LHS / Lyndon
\(H^p(B;H^q(F))\Rightarrow H^{p+q}(E)\)	Serre SS (fib.)
\(\operatorname{Tot}(C^{\bullet,\bullet})\), two filtrations	Double complex \(\Rightarrow\) SS
First quadrant: finite \(d_r\) hits	Edge maps exist

Formula	Name
\(L(\pi,s)\) automorphic \(L\)-function	Automorphic \(L\)
\(L(\rho,s)\) from Galois rep \(\rho:\mathrm{Gal}_K\to\mathrm{GL}_n\)	Galois \(L\)
Global Langlands: \(\pi\leftrightarrow\rho\) matching \(L\)-factors	Reciprocity (conj.)
Local Langlands: \(\operatorname{Irr}(G(K))\leftrightarrow\) Weil–Deligne reps	LLC
Modularity: elliptic curves \(/\mathbb{Q}\leftrightarrow\) wt 2 cusp forms	Wiles et al.
Functoriality: \(L\)-homomorphisms induce \(\pi\mapsto\pi'\)	Langlands functoriality
Geometric Langlands: \(\operatorname{Bun}_G\) Hecke eigensheaves \(\leftrightarrow\) loc. sys.	Geometric form
Trace formula \(\leftrightarrow\) spectral side of automorphic forms	Arthur–Selberg

Formula	Name
\(\omega\in\Omega^2(M)\) closed, nondeg.	Symplectic form
\(\iota_{X_H}\omega=-dH\) (or \(+\); sign conv.)	Hamiltonian vector field
\(\{f,g\}=\omega(X_f,X_g)\)	Poisson bracket
\(\mathcal{L}_{X_H}\omega=0\)	Hamiltonian flow preserves \(\omega\)
\(dH(X_H)=0\)	Energy conservation
\(\omega_0=\sum dq_i\wedge dp_i\) on \(T^*Q\)	Canonical / Liouville form setup
\(\mu:M\to\mathfrak{g}^*\), \(\langle d\mu_\xi,v\rangle=\omega(X_\xi,v)\)	Moment map
\(\mu^{-1}(0)/G\) (reg., free) symplectic	Marsden–Weinstein reduction
\(\omega^n/n!\)	Liouville volume
Lagrangian \(L\): $\omega	_L=0$, \(\dim L=\frac12\dim M\)
\(f^*\omega=\omega\)	Symplectomorphism
Darboux: locally \(\omega=\omega_0\)	Canonical coordinates

Formula	Name
\((\mathcal{A},\varphi)\) noncomm.\ prob.\ space	\(\varphi\) tracial state (often)
\(a_1,\ldots,a_n\) free if mixed free cumulants vanish	Freeness
\(R_a(z)=\sum\kappa_n(a)z^{n-1}\)	\(R\)-transform (free cumulants)
\(R_{a+b}=R_a+R_b\) when \(a,b\) free	Free additivity
\(G_a(z)=\varphi((z-a)^{-1})\)	Cauchy / Stieltjes transform
\(G_a^{\langle-1\rangle}(w)=1/w+R_a(w)\)	\(R\) via inverse of \(G\)
Semicircle: \(R(z)=z\) (std.); \(G=(z-\sqrt{z^2-4})/2\)	Free CLT law
Free Poisson / Marchenko–Pastur	Free compound Poisson
\(S\)-transform: mult. free convolution \(\boxtimes\)	Multiplicative freeness
\(\kappa_n\) free cumulants via Möbius on N-C partitions	Speicher combinatorial formula
Asymptotic freeness of independent Wigner/Haar	Voiculescu (RMT link)
\(\varphi(a^*a)\ge0\); \(\\|\cdot\\|_\varphi\)	\(L^2(\varphi)\) GNS setup