= Regularity structure
A <regularity structure> is an algebraic framework for local expansions that can include singular distributions as well as smooth <functions>. Its formal data are $(A,T,G)$: a locally finite set $A\subset\mathbb R$ containing zero and bounded below; a graded <vector space> $T=\bigoplus_{\alpha\in A}T_\alpha$ of <Banach spaces>, with $T_0$ the constants; and a group $G$ of linear maps fixing $1$ such that
$$
\Gamma\tau-\tau\in\bigoplus_{\beta<\alpha}T_\beta\qquad(\tau\in T_\alpha).
$$
Thus changing an expansion's base point modifies a symbol only through lower-homogeneity terms. <Martin Hairer> introduced this framework to treat classes of singular <stochastic partial differential equations>.<A theory of regularity structures by Martin Hairer>[Definition 2.1, arXiv v4 printed pages 18–19; Introduction, motivation from singular equations]{ref}
Ordinary polynomials provide a complete elementary example. In one dimension take $A=\{0,1,2,\ldots\}$, $T_n$ spanned by $X^n$, and realize $X^n$ around $x$ as $(z-x)^n$. The function $f(z)=z^2$ then has abstract local expansion $a_x=x^2+2xX+X^2$. To transfer the expansion from $y$ to $x$, substitute $X\mapsto X+x-y$. For example,
$$
\Gamma_{12}(4+4X+X^2)=4+4(X-1)+(X-1)^2=1+2X+X^2.
$$
The expression on the left is the expansion at $y=2$ transported to $x=1$. In particular, $\Gamma_{12}X^2-X^2=-2X+1$ has strictly lower degree, illustrating the defining triangular property.
A <model of a regularity structure> adds concrete distributions and analytic bounds to these abstract symbols. Writing their realizations as $\Pi_x$, compatibility requires $\Pi_x\Gamma_{xy}=\Pi_y$ and $\Gamma_{xy}\Gamma_{yz}=\Gamma_{xz}$. For singular equations the symbols extend beyond monomials; constructing a model, proving reconstruction and controlling corrections are additional steps. The triple alone does not make arbitrary products of distributions meaningful.<A theory of regularity structures by Martin Hairer>[Sections 2.2–2.3, polynomial example and Definition 2.17 of a model]{ref}
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