In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.

When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

Using properties of Appell–Lerch functions, we give insightful proofs for six of Ramanujan’s identities for the tenth-order mock theta functions.

We define the notion of $\unicode[STIX]{x1D6F7}$-Carleson measures, where $\unicode[STIX]{x1D6F7}$ is either a concave growth function or a convex growth function, and provide an equivalent definition. We then characterize $\unicode[STIX]{x1D6F7}$-Carleson measures for Bergman–Orlicz spaces and use them to characterize multipliers between Bergman–Orlicz spaces.

We construct combinatorial bases of the $T$-equivariant cohomology $H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)$ of the Bott–Samelson variety $\unicode[STIX]{x1D6F4}$ under some mild restrictions on the field of coefficients $k$. These bases allow us to prove the surjectivity of the restrictions $H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)\rightarrow H_{T}^{\bullet }(\unicode[STIX]{x1D70B}^{-1}(x),k)$ and $H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)\rightarrow H_{T}^{\bullet }(\unicode[STIX]{x1D6F4}\setminus \unicode[STIX]{x1D70B}^{-1}(x),k)$, where $\unicode[STIX]{x1D70B}:\unicode[STIX]{x1D6F4}\rightarrow G/B$ is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of $H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)$ and restricting them to $\unicode[STIX]{x1D70B}^{-1}(x)$ or $\unicode[STIX]{x1D6F4}\setminus \unicode[STIX]{x1D70B}^{-1}(x)$ respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image $\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{k}_{_{\unicode[STIX]{x1D6F4}}}$. This algorithm avoids division by 2, which allows us to re-establish 2-torsion for parity sheaves in Braden’s example, Braden and Williamson [‘Modular intersection cohomology complexes on flag varieties’, Math. Z.272(3–4) (2012), 697–727].

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.