In this article, we study a spectrum $K\left({\mathcal{V}}_k\right)$ such that ${\pi }_{0}K\left({\mathcal{V}}_k\right)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_0\left[{\mathcal{V}}_k\right]$ and to show that classes in the kernel of multiplication by $\left[{\mathbb{A}}^1\right]$ can always be represented as $\left[X\right]-\left[Y\right]$, where $\left[X\right]\ne \left[Y\right]$, $X\times {\mathbb{A}}^1$, and $Y\times {\mathbb{A}}^1$ are not piecewise-isomorphic, but $[X\times {\mathbb{A}}^1]=[Y\times {\mathbb{A}}^1]$ in $K_0\left[{\mathcal{V}}_k\right]$. Along the way, we present a new proof of the result of Larsen–Lunts on the structure on $K_0\left[{\mathcal{V}}_k\right]/\left(\right[{\mathbb{A}}^1\left]\right)$.

We prove that there exist uncountably many separable ${\mathrm{II}}_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are nonisomorphic. In fact, we prove that the families of nonisomorphic ${\mathrm{II}}_1$ factors originally introduced by McDuff are such examples. This entails the existence of a continuum of nonelementarily equivalent ${\mathrm{II}}_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

Let $F$ be a $p$-adic field, that is, a finite extension of ${\mathbb{Q}}_p$ for some prime $p$. The local Langlands correspondence (LLC) attaches to each continuous $n$-dimensional $\Phi$-semisimple representation $\rho$ of $W{\text{'}}_F$, the Weil–Deligne group for $\overline{F}/F$, an irreducible admissible representation $\pi \left(\rho \right)$ of ${\mathrm{GL}}_n\left(F\right)$ such that, among other things, the local $L$- and $\epsilon$-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this article, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square $\epsilon$-factors, that is, that $\epsilon (s,{\Lambda }^2\rho ,\psi )=\epsilon (s,\pi (\rho ),{\Lambda }^2,\psi )$ and $\epsilon (s,{Sym}^2\rho ,\psi )=\epsilon (s,\pi (\rho ),{Sym}^2,\psi )$. The agreement of the $L$-functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic $\gamma$-factor $\gamma (s,\pi ,{\Lambda }^2,\psi )$ under highly ramified twists when $\pi$ is supercuspidal. This last step is achieved by relating the $\gamma$-factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation $\pi$ then follows from those of the corresponding arithmetic $\gamma$-factors as a corollary.

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait’s conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and of Hirasawa and Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juhász and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.

We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative $L^p$-spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group ${\mathbf{F}}_{\infty }$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu’s amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness problem posed by Ozawa, a length-independent estimate for Junge–Parcet–Xu’s free Rosenthal’s inequality, a Littlewood–Paley–Stein-type inequality for geodesic paths of free groups, and a length reduction formula for $L^p$-norms of free group von Neumann algebras.