A conjecture of Mináč and Tân predicts that for any $n\ge 3$, any prime p, and any field k, the Massey product of n Galois cohomology classes in $H^1(k,\mathbf{Z}∕p\mathbf{Z})$ must vanish if it is defined. We establish this conjecture when k is a number field.

This work proves rigorous results about the vanishing mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size $\mathit{\epsilon }\downarrow 0$ to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitté in 2001 and predicted heuristically before in works of Allaire, Kohn, and Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the nonconvexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, settling this open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) $\mathrm{div}$-quasiconvex quadratic forms, computations involving the Hashin–Shtrikman bounds for the Kohn–Strang integrand, and the characterization of limit minimizers due to Bouchitté and Buttazzo.

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If $\mathrm{char}(\mathbf{k})=p>0$, then we use this method to construct the incompressible abelian symmetric tensor categories ${\mathrm{Ver}}_{p^n}$, ${\mathrm{Ver}}_{p^n}^{+}$ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for $n=1$, and by Benson–Etingof for $p=2$. Namely, ${\mathrm{Ver}}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for ${\mathit{SL}}_2(\mathbf{k})$ by the nth Steinberg module, and ${\mathrm{Ver}}_{p^n}^{+}$ is its subcategory generated by ${\mathit{PGL}}_2(\mathbf{k})$-modules. We show that ${\mathrm{Ver}}_{p^n}$ are reductions to characteristic p of Verlinde braided tensor categories in characteristic 0, which explains the notation. We study the structure of these categories in detail and, in particular, show that they categorify the real cyclotomic rings $\mathbb{Z}[2cos(2\mathrm{\pi }∕p^n)]$, and that ${\mathrm{Ver}}_{p^n}$ embeds into ${\mathrm{Ver}}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union ${\mathrm{Ver}}_{p^{\mathrm{\infty }}}$ of the nested sequence ${\mathrm{Ver}}_p\subset {\mathrm{Ver}}_{p^2}\subset \cdots$ . This would provide an analogue of Deligne’s theorem in characteristic 0 and a generalization of the results of Coulembier, Etingof, and Ostrik, which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and, moreover, the fiber functor lands in ${\mathrm{Ver}}_p$ (in the case of fusion categories, this was shown earlier by Ostrik). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories ${\mathrm{Ver}}_{p^n}$.