We estimate the sums $${\sum }_{n\le x}\Lambda \left(n\right)f\left(n\right)g\left(n\right)exp\left(2i\pi \vartheta n\right)$$ and $${\sum }_{n\le x}\mu \left(n\right)f\left(n\right)g\left(n\right)exp\left(2i\pi \vartheta n\right),$$ where $\Lambda$ denotes the von Mangoldt function (and $\mu$ the Möbius function) whenever $q_1$ and $q_2$ are two coprime bases, $f$ (resp., $g$) is a strongly $q_1$-multiplicative (resp., strongly $q_2$-multiplicative) function of modulus $1$, and $\vartheta$ is a real number. The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation, and combinatorial arguments. We deduce from these estimates a prime number theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.

We use inequalities to completely describe the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of Lis to give a simple bijection between such correlation matrices and points in the totally nonnegative part of the orthogonal Grassmannian, which was introduced in 2013 in the study of the scattering amplitudes of Aharony–Bergman–Jafferis–Maldacena (ABJM) theory. We also show that the edge parameters of the Ising model for reduced networks can be uniquely recovered from boundary correlations, solving the inverse problem. Under our correspondence, the Kramers–Wannier high/low temperature duality transforms into the cyclic symmetry of the Grassmannian, and using this cyclic symmetry, we prove that the spaces under consideration are homeomorphic to closed balls.

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y\to Y_{\left(0\right)}$ which is well understood. Early on it was found that the induced map $[X,Y]\to [X,Y_{\left(0\right)}]$ on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that, as the complexity (in a suitable sense) of a rational mapping class increases, these sizes are at most polynomial. This “torsion” information about $[X,Y]$ is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of $Y$ in at least some cases. The notion of complexity is geometric, and we also prove a conjecture of Gromov regarding the number of mapping classes that have Lipschitz constant at most $L$.

We explore the structure of the Hitchin morphism for higher-dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a nonlinear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphism for higher-dimensional varieties, the invariant theory of the commuting schemes, and Weyl’s polarization theorem. We use the factorization of the Hitchin morphism to construct the spectral and cameral covers. In the case of general linear groups and algebraic surfaces, we show that spectral surfaces admit canonical finite Cohen–Macaulayfications, which we call the Cohen–Macaulay spectral surfaces, and we use them to obtain a description of the generic fibers of the Hitchin morphism similar to the case of curves. Finally, we study the Hitchin morphism for some classes of algebraic surfaces.