We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat–Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of ${Spin}_8$ (ramified triality) provides an input needed in the article by He–Pappas–Rapoport classifying Shimura varieties with good or semi-stable reduction.

A group $G$ is periodic of bounded exponent if there exists $k\in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every finitely generated periodic group of bounded exponent $G<{Diff}_{\omega }\left({\mathbb{S}}^2\right)$ is finite, where ${Diff}_{\omega }\left({\mathbb{S}}^2\right)$ denotes the group of diffeomorphisms of ${\mathbb{S}}^2$ that preserve an area form $\omega$.

We prove a nonlinear variant of the general Brascamp–Lieb inequality. Our proof consists of running an efficient, or “tight,” induction-on-scales argument, which uses the existence of Gaussian near-extremizers to the underlying linear Brascamp–Lieb inequality (Lieb’s theorem) in a fundamental way. A key ingredient is an effective version of Lieb’s theorem, which we establish via a careful analysis of near-minimizers of weighted sums of exponential functions. Instances of this inequality are quite prevalent in mathematics, and we illustrate this with some applications in harmonic analysis.

We introduce an explicit description of the ${\Phi }_3^4$ measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale regularization parameter. Once small scale singularities have been renormalized by the standard counterterms, $\Gamma$-convergence allows us to extend the variational characterization to the unregularized model.