Let $g_0,\dots ,g_k:\mathbb{N}\to \mathbb{D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots ,h_k\in \mathbb{Z}$ be shifts. We consider correlation sequences $f:\mathbb{N}\to \mathbb{Z}$ of the form $$f(a):=\underset{m\to \infty }{}\frac{1}{log{\omega }_m}{\sum }_{x_m/{\omega }_m\le n\le x_m}\frac{g_0(n+a{h}_0)\cdots g_k(n+a{h}_k)}{n},$$ where $1\le {\omega }_m\le x_m$ are numbers going to infinity as $m\to \infty$ and $$ is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore, if the multiplicative function $g_0\cdots g_k$ “weakly pretends” to be a Dirichlet character $\chi$, the periodic functions $f_i$ can be chosen to be $\chi$-isotypic in the sense that $f_i\left(ab\right)=f_i\left(a\right)\chi \left(b\right)$ whenever $b$ is coprime to the periods of $f_i$ and $\chi$, while if $g_0\cdots g_k$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.

We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler–Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension $3$ of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of $3$-dimensional canonical and terminal singularities, which was established during the study of the explicit $3$-dimensional minimal model program.

We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in ${\mathbb{R}}^n$, $1\le d\le n-1$. We show that a canonical equiaffine-invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of Oberlin with an exponent which is best possible. The proof combines aspects of geometric invariant theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

Kuznetsov has conjectured that Pfaffian varieties should admit noncommutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by Hori.