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Mathematics
$$\int_a^b f'(x),円 dx = f(b) - f(a) \qquad\qquad \frac{d}{dx}\!\left(\int_a^x f(t),円 dt\right) = f(x)$$

The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.

Terence Tao, Differential Forms and Integration
Differential Forms
$$\int_{\partial M}\omega=\int_M d\omega, \qquad d^2=0$$

Differential forms turn orientation, boundary, and change of variables into one calculus: the exterior derivative records what a form contributes on the boundary.

Čech Cohomology
$$ C^q(\mathfrak U,\mathcal F)= \prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0\cdots i_q}), \qquad \check H^q=\ker\delta/\operatorname{im}\delta $$

Čech cohomology is the computational face of sheaf cohomology: write sections on overlaps, apply the alternating coboundary, and keep the classes that cannot be patched away.

Sheaf Cohomology
$0ドル\to\mathcal F'\to\mathcal F\to\mathcal F''\to0 \quad\Longrightarrow\quad H^q(X,\mathcal F)\to H^q(X,\mathcal F'')\to H^{q+1}(X,\mathcal F')$$

Sheaf cohomology measures the precise obstruction to turning compatible local analytic data into global geometric objects.

Ring Theory
$$A \longmapsto \operatorname{Spec} A,\qquad \mathcal O_{\operatorname{Spec} A}(D(f))=A_f,\qquad \widetilde M(D(f))=M_f$$

Prime ideals are the points, localization is restriction to an open set, and sheaves are the mechanism that turns local algebra into geometry.

Riemann-Roch
$$ \begin{aligned} \ell(D)-\ell(K-D) &= \deg D + 1 - g, \\ & \\ \chi\!\left(X,\mathcal O_X(D)\right) &= h^0\!\left(X,\mathcal O_X(D)\right) - h^1\!\left(X,\mathcal O_X(D)\right) %\\ % &= \deg D + 1 - g,\\ \end{aligned} $$

A divisor records the requested zeros and allowed poles; Riemann-Roch turns that data, together with the genus, into an exact dimension count.

Provable Security
$$\operatorname{Adv}^{\mathsf{scheme}}_{A}(\lambda)\le q(\lambda)\operatorname{Adv}^{\mathsf{assumption}}_{B}(\lambda)+\varepsilon(\lambda)$$

A proof in cryptography is a reduction: it states which attack game is being ruled out, which assumption is being invoked, and how much quantitative security is lost in the translation.

Coding Theory
$$C_L(D,G)=\{(f(P_1),\ldots,f(P_n)): f\in L(G)\}, \qquad C_\Omega(D,G)=C_L(D,G)^\perp$$

Error-correcting codes become geometric when codewords are evaluations of functions and parity checks are residues of differentials.

Bignum Arithmetic
$$ \begin{aligned} a &= \sum_{i=0}^{n-1} a_iB^i,\qquad b = \sum_{i=0}^{n-1} b_iB^i,\qquad B=2^{32} % ,\\ 0 &\le a_i,b_i < B, % \\ \operatorname{mul}_m(a,b) &= (a\cdot b)\bmod m \end{aligned} $$

A cryptographic bignum primitive is a proof-carrying C program: every word operation must preserve the integer invariant and the leakage boundary.

Elliptic Arithmetic
$$E/\mathbb F_p:\ y^2=x^3+ax+b,\qquad [k]P=P+\cdots+P$$

Elliptic-curve arithmetic is the group-law layer above bignum field arithmetic: words implement residues, residues implement points, and points implement public-key primitives.

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