We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph that does not contain $k\,+\,1$ vertex-disjoint triangles.

This extends a result of Moon [Canad. J. Math. 20 (1968), 96–102], which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi–Hajnal Theorem.

We prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a ${{C}^{*}}$-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow.

Erdős conjectured that for any set $A\,\subseteq \,\mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,\,C\,\subseteq \,\mathbb{N}$ such that $B\,+\,C\,\subseteq \,A$. We verify Erdős’ conjecture in the case where $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\,\subseteq \,\mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,\,C\,\subseteq \,\mathbb{N}$ such that $B\,+\,C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

Let $\theta \,\in \,\left[ 0,\,1 \right]$ be any irrational number. It is shown that the extended rotation algebra ${{B}_{\theta }}$ introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1–71, is always an $\text{AF}$ algebra.

Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta $ of $B$ is a certain Banach algebra product $A\,{{\times }_{\theta }}\,B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\,{{\times }_{\theta }}\,B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner–Schoenberg–Eberlein property and the algebra of Bochner–Schoenberg–Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\,{{\times }_{\theta }}\,B$.

Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.

Let $\beta:\,{{S}^{2n+1}}\,\to \,{{S}^{2n+1}}$ be a minimal homeomorphism $\left( n\,\ge \,1 \right)$. We show that the crossed product $C\left( {{S}^{2n+1}} \right)\,{{\rtimes }_{\beta}}\mathbb{Z}$ has rational tracial rank at most one. Let $\Omega $ be a connected, compact, metric space with finite covering dimension and with ${{H}^{1}}\left( \Omega ,\,\mathbb{Z} \right)\,=\,\left\{ 0 \right\}$. Suppose that ${{K}_{i}}\left( C\left( \Omega \right) \right)\,=\,\mathbb{Z}\,\oplus \,{{G}_{i}}$, where ${{G}_{i}}$ is a finite abelian group, $i\,=\,0,\,1$. Let $\beta :\,\Omega \,\to \,\Omega $ be a minimal homeomorphism. We also show that $A\,=\,C\left( \Omega \right)\,{{\rtimes }_{\beta}}\,\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\,\times \,\Omega $, where $X$ is the Cantor set.

We construct one-parameter families of overconvergent Siegel–Hilbert modular forms. This result has applications to the construction of Galois representations for automorphic forms of non-cohomological weights.

This paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space ${{\mathbb{P}}^{n}}$ with $n\,\ge \,3$. For each given point of ${{\mathbb{P}}^{n}}$ there is a subgroup of the entire Cremona group of dimension $n$ consisting of such maps. We study both geometric and group-theoretical properties of this notion. In the case where $n\,=\,3$ we describe an explicit set of generators of the group and give a homological characterization of a basic subgroup thereof.

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ${{\mathbb{C}}^{n}}$.