R. Gow proved that the order of a solvable rational group is divisible only by the primes 2, 3 and 5. In this paper it is proved that in a solvable rational group the Sylow 5-subgroup is always normal and elementary Abelian. Moreover, the structure of rational {2, 5}-groups is described in detail.

We study the inhomogeneous conormal derivative problem for the divergence form elliptic equation, assuming that the principal coefficients belong to the BMO space with small BMO semi-norms and that the boundary is $\delta$-Reifenberg flat. These conditions for the $W^{1, p}$-theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domain. In fact, the Reifenberg flatness is the minimal regularity condition for the $W^{1, p}$-theory.

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfaces in a four-dimensional Kähler manifold which are critical points of the area functional for Hamiltonian infinitesimal deformations. In this paper we analyze these surfaces in the complex projective plane: in a previous work we showed that they correspond locally to solutions to an integrable system, formulated as a zero curvature on a (twisted) loop group. Here we give an alternative formulation, using non-twisted loop groups and, as an application, we show in detail why Hamiltonian stationary Lagrangian tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.

In this paper, we study applications of Atiyah–Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel. For example, we compute the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky. The cobordism ring $MGL / 2^{*,*} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz. Moreover, we study algebraic cobordism of the classifying spaces $BG$ for algebraic groups over $\mathbb{C}$. For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.

It is shown that the classical John ellipsoid, the Petty ellipsoid and a recently discovered ‘dual’ of the Legendre ellipsoid are all special cases ($p = \infty$, $p = 1$ and $p = 2$) of a family of $L_p$ ellipsoids which can be associated with a fixed convex body. This insight allows for a unified view of, alternative approaches to, and extensions of some basic results in convex geometry.

Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson–Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.