We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

In the mid-1960s Borevi$\setminus$v\{c\} and Faddeev initiated the study of the Galois module structure of groups of \$p\$th-power classes of cyclic extensions \$K/F\$ of \$p\$th-power degree. They determined the structure of these modules in the case when \$F\$ is a local field. In this paper we determine these Galois modules for all base fields \$F\$.

This corrigendum supplies different proofs for those places in the original paper where a certain theorem by Rickard was used.

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

We employ techniques from the theory of disintegration of measures to study the Lifting Problem for commuting $n$-tuples of subnormal weighted shifts. We obtain a new necessary condition for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal $2$-variable weighted shifts, we then find the precise relation between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights.

We present a vector field method for showing that certain subgroups of the mapping class group $\Gamma$ of a Riemann surface of infinite topological type act properly discontinuously. We apply the method to the group of homotopy classes of quasiconformal self-maps of the complement $\Omega$ of a Cantor set in $\mathbb{C}$. When the Cantor set has bounded geometric type, we show that $\Gamma(\Omega)$ acts on the Teichmüller space $T(\Omega)$ properly discontinuously. Also, we apply the same method to show that the pure mapping class group $\Gamma_0(\Omega \cup \{\infty\})$ acts properly discontinuously on $T(\Omega \cup \{\infty\})$.

In previous joint work with Cappell and Shaneson, we have established an Atiyah–Lusztig–Meyer-type multiplicative characteristic class formula for the twisted signature and, more generally, the twisted $L$-class, of a stratified Witt space. The present paper shows that these formulae hold even when the stratified space does not satisfy the Witt condition. It constitutes one of the first applications of signature homology.

In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite-type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study {\it first order semi-local invariants} of stable maps and those of stable fold maps of a closed orientable 3-dimensional manifold into the plane. We show that there are essentially eight first order semi-local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi-local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps.

This paper constructs hyper-homology spectral sequences of $\mathbb{Z}$-graded and $RO_{G}$-graded Mackey functors that compute $\mathrm{Ext}$ and $\mathrm{Tor}$ over $G$-equivariant $S$-algebras ($A_{\infty}$ ring spectra) for finite groups $G$. These specialize to universal coefficient and K\"unneth spectral sequences.