The analyst’s traveling salesman problem (TSP) is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ${\ell }_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul’s necessary and sufficient conditions for a bounded set $E\subset {\ell }_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\mathrm{\infty }$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\ne 2$. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the analyst’s TSP in general metric spaces.

We prove that in any Banach space, the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I (also published in this issue), this completes the proof of the necessary half of the analyst’s traveling salesman theorem with sharp exponent in uniformly convex spaces.

Newman showed that for primes $p\ge 5$, an integral circulant determinant of prime power order $p^t$ cannot take the value $p^{t+1}$ once $t\ge 2$. We show that many other values are also excluded. In particular, we show that $p^{2t}$ is the smallest power of p attained for any $t\ge 3$, $p\ge 3$. We demonstrate the complexity involved by giving a complete description of the $25\times 25$ and $27\times 27$ integral circulant determinants. The former case involves a partition of the primes that are $1mod5$ into two sets, Tanner’s artiads and perissads, which were later characterized by E. Lehmer.

We prove that every nondegenerate Banach space representation of the Drinfeld–Jimbo algebra $U_q(\mathfrak{g})$ of a semisimple complex Lie algebra $\mathfrak{g}$ is finite dimensional when $|q|\ne 1$. As a corollary, we find an explicit form of the Arens–Michael envelope of $U_q(\mathfrak{g})$, which is similar to that of $U(\mathfrak{g})$ obtained by Joseph Taylor in 1970s. In the case when $\mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_2$, we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra $\tilde{U}{(\mathfrak{s}{\mathfrak{l}}_2)}_{\hslash }$ (with $e^{\hslash }=q$) and show that it is simpler than for $U_q(\mathfrak{s}{\mathfrak{l}}_2)$. For example, all irreducible continuous representations of $\tilde{U}{(\mathfrak{s}{\mathfrak{l}}_2)}_{\hslash }$ are finite dimensional for every admissible value of the complex parameter ℏ, while $U_q(\mathfrak{s}{\mathfrak{l}}_2)$ has a topologically irreducible infinite-dimensional representation when $|q|=1$ and q is not a root of unity.

In this article, we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1{\otimes }_R{S}_2$, where $S_1$, $S_2$ are Tor-independent over R. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1{\otimes }_R{S}_2$, denoted $\mathrm{\pi }(S)$, in terms of those of R, $S_1$, and $S_2$; namely, $\mathrm{\pi }(S)$ is the pullback of (adjusted) Lie algebras along the maps $\mathrm{\pi }(S_i)\to \mathrm{\pi }(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André–Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of R, $S_1$, $S_2$, and S.

Dowdall and Taylor observed that, given a finite-index subgroup of a free group, taking covers induces an embedding from the outer space of the free group to the outer space of the subgroup, this embedding is an isometry with respect to the (asymmetric) Lipschitz metric and that the embedding sends folding paths to folding paths. The purpose of this note is to extend this result to virtually free groups. We further extend a result of Francaviglia and Martino, proving the existence of “candidates” for the Lipschitz distance between points in the outer space of the virtually free group. Additionally, we identify a deformation retraction of the spine of the outer space for the virtually free group with the space considered by Krstić and Vogtmann.