We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for $\mathbb{Q}$-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, let $\omega$ be a positive Musielak–Orlicz function on $(0,\infty )$ of uniformly strictly critical lower-type $p_{\omega }\in (0,1]$, and let $\rho (x,t)=t^{-1}/{\omega }^{-1}(x,t^{-1})$ for $x\in {\mathbb{R}}^n$, $t\in (0,\infty )$. In this paper, we study the Musielak–Orlicz Hardy space $H_{\omega ,L}\left({\mathbb{R}}^n\right)$ and its dual space ${\mathrm{BMO}}_{\rho ,L^{*}}\left({\mathbb{R}}^n\right)$, where $L^{*}$ denotes the adjoint operator of $L$ in $L^2\left({\mathbb{R}}^n\right)$. The $\rho$-Carleson measure characterization and the John–Nirenberg inequality for the space ${\mathrm{BMO}}_{\rho ,L}\left({\mathbb{R}}^n\right)$ are also established. Finally, as applications, we show that the Riesz transform $\nabla L^{-1/2}$ and the Littlewood–Paley $g$-function $g_L$ map $H_{\omega ,L}\left({\mathbb{R}}^n\right)$ continuously into $L\left(\omega \right)$.

Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and co-t-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.

The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal and when it is Gorenstein. In addition, Gröbner bases of toric ideals of centrally symmetric configurations are discussed. Special attention is given to centrally symmetric configurations of unimodular matrices and to those of incidence matrices of finite graphs.