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 show that if a unital injective endomorphism of a $C^{\ast }$ -algebra admits a transfer operator, then both of them are compressions of mutually inverse automorphisms of a bigger algebra. More generally, every interaction group – in the sense of Exel – extending an action of an Ore semigroup by injective unital endomorphisms of a $C^{\ast }$ -algebra, admits a dilation to an action of the corresponding enveloping group on another unital $C^{\ast }$ -algebra, of which the former is a $C^{\ast }$ -subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed, and it is faithful if and only if the interaction group is also faithful.

Inspired by the recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki, we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi–Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.

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.

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$ . We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviours (locally Gaussian and at infinity sub-Gaussian), in which case the previous theory does not apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the $H^{p}$ space corresponding to Gaussian estimates may not coincide with $L^{p}$ . As a motivation for this theory, we show that the Riesz transform maps our Hardy space $H^{1}$ into $L^{1}$ .

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Let $a\in \mathbb{R}$ , and let $k(a)$ be the largest constant such that $\sup |\text{cos}(na)-\cos (nb)|<k(a)$ for $b\in \mathbb{R}$ implies that $b\in \pm a+2\unicode[STIX]{x1D70B}\mathbb{Z}$ . We show that if a cosine sequence $(C(n))_{n\in \mathbb{Z}}$ with values in a Banach algebra $A$ satisfies $\sup _{n\geq 1}\Vert C(n)-\cos (na).1_{A}\Vert <k(a)$ , then $C(n)=\cos (na).1_{A}$ for $n\in \mathbb{Z}$ . Since $\!\sqrt{5}/2\leq k(a)\leq 8/3\!\sqrt{3}$ for every $a\in \mathbb{R}$ , this shows that if some cosine family $(C(g))_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies $\sup _{g\in G}\Vert C(g)-c(g)\Vert <\!\sqrt{5}/2$ for some scalar cosine family $(c(g))_{g\in G}$ , then $C(g)=c(g)$ for $g\in G$ , and the constant $\!\sqrt{5}/2$ is optimal. We also describe the set of all real numbers $a\in [0,\unicode[STIX]{x1D70B}]$ satisfying $k(a)\leq \frac{3}{2}$ .

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.

Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph–edge pairs have this property. In particular, we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series–parallel graphs with the property.

Given a partial action $\unicode[STIX]{x1D703}$ of a group on a set with an algebraic structure, we construct a reflector of $\unicode[STIX]{x1D703}$ in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if $\unicode[STIX]{x1D703}$ is a partial action on an algebra from a variety $\mathsf{V}$ , then we show that the problem reduces to the embeddability of a certain generalized amalgam of $\mathsf{V}$ -algebras associated with $\unicode[STIX]{x1D703}$ . As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.

We study the boundedness from $H^{p(\cdot )}(\mathbb{R}^{n})$ into $L^{q(\cdot )}(\mathbb{R}^{n})$ of certain generalized Riesz potentials and the boundedness from $H^{p(\cdot )}(\mathbb{R}^{n})$ into $H^{q(\cdot )}(\mathbb{R}^{n})$ of the Riesz potential, both results are achieved via the finite atomic decomposition developed in Cruz-Uribe and Wang [‘Variable Hardy spaces’, Indiana University Mathematics Journal63(2) (2014), 447–493].

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$ . Our main results represent improvements as well as natural extensions of many previously known results.

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].

We discuss the internal structure of graph products of right LCM semigroups and prove that there is an abundance of examples without property (AR). Thereby we provide the first examples of right LCM semigroups lacking this seemingly common feature. The results are particularly sharp for right-angled Artin monoids.

The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff and totally disconnected unit space an $R$ -algebra $R\,\mathscr{G}$ . In this paper we characterize when $R\,\mathscr{G}$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

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.