We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\ge 2$. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base dimension under the condition that all the Hodge numbers of type $h^{0,p}(X)$ are derived invariant.

We explore transformation groups of manifolds of the form $M\times S^n$, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action of a finite group. In particular, we prove that for $n=2$, there exists an infinite family of distinct nondiagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on $M\times S^1$, where M belongs to the class of “almost asymmetric” manifolds considered previously by Puppe and Kreck.

We classify the blocks and compute the Verma flags of tilting and projective modules in the BGG category $\mathcal{O}$ for the exceptional Lie superalgebra $G(3)$. The projective injective modules in $\mathcal{O}$ are classified. We also compute the Jordan–Hölder multiplicities of the Verma modules in $\mathcal{O}$.

Assume that G is a chordal graph with edge ideal $I(G)$ and induced matching number $\mathrm{\nu }(G)$. We denote the sth symbolic power of $I(G)$ by $I{(G)}^{(s)}$. It is shown that $\mathrm{reg}(I{(G)}^{(s)})=2s+\mathrm{\nu }(G)-1$ for every integer $s\ge 1$.

A formula on Stirling numbers of the second kind $S(n,k)$ is proved. As a corollary, for odd n and even k, it is shown that $k!S(n,k)$ is a positive multiple of the greatest common divisor of $j!S(n,j)$ for $k+1\le j\le n$. Also, as an application to algebraic topology, some isomorphisms of unstable $K^1$-groups of stunted complex projective spaces are deduced.

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Bégueri, Bester, and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups.

We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm which we used to obtain first examples of so-called noneasy linear categories of partitions. All of the examples that we constructed are proved to be indeed new and noneasy. We interpret some of the new categories in terms of quantum group anticommutative twists.