Foundations

For the heat equation u-t−kappa Delta-u=f with zero Dirichlet data, testing by u yields a balance between the time derivative of half the squared L2 norm, gradient dissipation and forcing.

Over an algebraically closed field k, polynomial zero sets V(I) have coordinate ring k[x1,...,xn]/I(V(I)).

An affine scheme Spec(R) pairs the space of prime ideals with a structure sheaf.

A ring is reduced when it has no nonzero nilpotents.

The fundamental group pi1(X,x0) consists of based loops modulo homotopies fixing endpoints, with concatenation.

A chain complex has boundary maps with boundary²=0, and homology is cycles modulo boundaries.

For smooth complex algebraic varieties, analytic topology, algebraic coordinate structure and smooth de Rham theory offer complementary invariants.

An unbounded operator is specified by a linear domain D(T) in a Hilbert space and a map T:D(T)→H.

For a closed densely defined symmetric operator T, deficiency indices n-plus and n-minus are the dimensions of ker(T*−iI) and ker(T*+iI).

A self-adjoint operator has a projection-valued spectral measure E on R and can be represented as an integral of lambda dE(lambda).

If P is a constant-coefficient elliptic differential operator of order m on R-n and u is a distribution on an open U, P-u in local H-s implies u in local H-(s+m).

A smooth elliptic operator on a compact manifold without boundary maps H^(s+m) to H-s with finite-dimensional kernel and cokernel and closed range.

For incompressible viscous flow, divergence-free initial data and pressure enforce the constraint in u-t+(u·grad)u=nu Delta-u−grad p+f.

A category has objects, composable morphisms, associative composition and an identity for every object.

A natural transformation eta:F⇒G assigns a morphism eta-X:F(X)→G(X) to every object so that G(f) eta-X=eta-Y F(f).

For a locally small category C, a functor F:C→Set and object c, natural transformations from Hom(c,−) to F correspond bijectively to elements of F(c), by evaluating at the identity of c.

An adjunction F left-adjoint to U is a natural bijection Hom-D(FX,Y)≅Hom-C(X,UY).

In intensional dependent type theory, a term x:A inhabits a type, dependent products encode functions/proofs with variable-dependent result types, and identity types encode identifications.

Univalence states that the canonical map from identity of types in a universe to equivalence of types is itself an equivalence.

For Re(s)>1, zeta(s)=sum n^(−s)=product over primes (1−p^(−s))^(−1), with absolute convergence.

The completed function xi(s)=one-half s(s−1) pi^(−s/2) Gamma(s/2) zeta(s) extends to an entire function with xi(s)=xi(1−s).

A character modulo q is a multiplicative character of the units modulo q extended by zero on integers not coprime to q.

The prime-counting function satisfies pi(x)~x/log x.

The Riemann hypothesis asserts that every nontrivial zero of zeta has real part one-half.

A finite Dirichlet polynomial D-N(t)=sum from N to 2N of b-n n^(it), with bounded coefficients, converts arithmetic questions into estimates for the set of t where |D-N(t)| is large.

Guth and Maynard's work, published in Annals of Mathematics on 1 March 2026, improves large-value estimates near the N^(3/4) regime.

For n integrable Hamiltonian degrees of freedom on a regular compact connected common level set of n independent commuting integrals, local canonical actions I_i and angles θ_i describe torus motion.

For H=H₀(I)+εH₁(I,θ), an integer resonance k·ω(I)=0 makes that Fourier mode unsuitable for ordinary nonresonant elimination.

Independent second-class constraints φ_a=0 have an invertible Poisson matrix C_ab={φ_a,φ_b}.

For separable H=pᵀM⁻¹p/2+V(q), fixed-step kick–drift–kick Verlet is second order, reversible and symplectic.