The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces $G/H$ in the case where $G$ is of type $E_8$. In the present article, we give the explicit form of automorphisms $\tilde{w}_4$, $\tilde{\upsilon}_4$ and $\tilde{\mu}_4$ of order four on $E_8$ induced by the $C$-linear transformations $w_4, \upsilon_4$ and $\mu_4$ of the 248-dimensional vector space $𝔢^C_8$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_4}, (E_8)^{\upsilon_4}$ and $(E_8)^{\mu_4}$ of $E_8$. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $G/H$ above corresponding to the Lie algebras $𝔥 = i \boldsymbol{R} \oplus 𝔰𝔲(8)$, $i\boldsymbol{R} \oplus 𝔢_7$ and $𝔥 = 𝔰𝔲(2) \oplus 𝔰𝔲(8)$, where $𝔥 = \mathrm{Lie}(H)$.

We consider the Grassmann manifold of associative subspaces in the space of imaginary octonions, which we call the associative Grassmann manifold. It is known that the associative Grassmann manifold is an eight-dimensional compact symmetric quaternionic Kähler manifold. We construct interesting examples of four-dimensional complex submanifolds of the associative Grassmann manifold.

We define a parabolic flow of almost Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. We show that the flow has a unique shorttime solution and also show a stability result when the background metric is quasi-Kähler with constant scalar curvature.

Let $C$ be a curve on the weighted projective plane $\mathbf{P}(1, 1, 4)$ which is of Fermat type or almost Fermat type. We construct K3 surfaces which are double covers of $\mathbf{P}(1, 1, 4)$ and which contain pointed curves with symmetric Weierstrass semigroups which are double covers of $C$.

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $(\Gamma, \boldsymbol{v})$ having a fixed oriented modulation $(𝔐, \Omega)$ over the ground field $K$. Here $K$ is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces.

Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting $\Lambda$-module, where $\Lambda$ is the tensor $K$-algebra of $(𝔐, \Omega)$. We count the number of isomorphism classes of hom-orthogonal partial tilting $\Lambda$-modules of type $\mathbf{B}_n$, $\mathbf{C}_n$, $\mathbf{F}_4$ and $\mathbf{G}_2$. As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.