Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same structure group $\mathrm{G}$ captures different classes of equations; depending on the class, different (sub)spaces $\mathrm{T}$ matter, which correspond to linear combinations of different types of trees.

In our approach to $\mathrm{G}$, we start from the space of tuples $(a,p)$ of (polynomial) nonlinearities $a$ and space-time polynomials $p$, which we think of parameterizing the entire manifold of solutions $u$ (satisfying the equation up to space-time polynomials) via re-centering. We consider the actions of a shift by a space-time vector $h\in \mathbb{R}^{d+1}$ and of tilt by space-time polynomial $q$ on $(a,p)$-space, where, crucially, the tilt by a constant is treated as a shift of the (one-dimensional) $u$-space. We consider the infinitesimal generators of these actions, and pull them back as derivations on the algebra of formal power series $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ in the natural coordinates $\{\mathrm{z}_{k}\}_{k\in \mathbb{N}_{0}}$ and $\{\mathrm{z}_{\mathrm{n}}\}_{\mathrm{n}\in \mathbb{N}_{0}^{d+1}-\{\mathrm{O}\}}$ of $(a,\ p)$-space. This defines a Lie algebra $\mathrm{L}\subset \mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]])$ . Loosely speaking, the corresponding Lie group coincides with $\mathrm{G}^{*}$, but we follow an algebraic path to construct G.

As a group, $\mathrm{G}\subset(\mathrm{T}^{+})^{*}$ arises in the standard way from the Hopf algebra $\mathrm{T}^{+}$ that is obtained from dualizing the universal enveloping algebra $\mathrm{U}(\mathrm{L})$ . Here, gradedness and finiteness properties are needed for the well-posedness of the co-product $\triangle^{+}:\mathrm{T}^{+}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}^{+}$ and the antipode. The passage from $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ to a smaller (linear) subspace $\mathrm{T}^{*}$ is needed for dualizing the module defined through $\mathrm{L}\subset$ End(T$*$) to obtain the co-module $\triangle:\mathrm{T}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}$. This yields the representation $\mathrm{G}\subset$ End(T). Both $\triangle$ and $\triangle^{+}$ satisfy the postulates of regularity structures, in particular the properties that intertwine $\triangle, \triangle^{+}$, and the family of re-centering maps $\mathcal{J}_{\mathrm{n}}:\mathrm{T}\rightarrow \mathrm{T}^{+}$. The latter relies on choosing a natural basis of $\mathrm{U}(\mathrm{L})$ , different from the standard Poincar\'{e}-Birkhoff-Witt basis, for dualization.

This is joint work with P. Linares and M. Tempelmayr.[-]

We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic $\Phi _{d}^{3}$ model.[-]