This note gives examples of nonconvex self-similar solutions for a crystalline curvature flow with an interfacial energy of which the Wulff shape is a regular triangle or a square.

Let $\widetilde{E}_8$ be the 3-connected covering space of the 1-connected, compact exceptional group $E_8$, which is regarded as the loop space of the homotopy fibre $B\widetilde{E}_8$ of a map from $BE_8$, the classifying space of $E_8$, to an Eilenberg-MacLane space. The Stiefel-Whitney classes of the adjoint representation of $E_8$ induce elements of the mod 2 cohomology of $B\widetilde{E}_8$. These images are computed.

We construct an elliptic curve with non-constant $j$-invariant of rank $\ge 5$ with a rational point of order 4 over $\mathcal{Q}(t)$.

In this article we shall investigate Hardy-Sobolev inequalities and improve them by adding a term with a singular weight of the type $(\log(1/|x|))^{-2}$. We show that this weight function is optimal in the sense that the inequality fails for any other weight more singular than this one. As an application, we use our improved inequality to study the weighted eigenvalue problem stated in §5.

We prove the following result that answers a question of M. Chen: Let $X$ be a Gorenstein minimal complex projective 3-fold of general type with locally factorial terminal singularities. If $|K_X|$ defines a generically finite map $\phi\colon X \dashrightarrow \mathbf{P}^{p_g-1}$, then $\deg(\phi) \leq 576$. For any positive integer $m > 0$, we give infinitely many examples of (non-Gorenstein) 3-folds of general type with canonical map of degree $m$.