We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in ${\mathbb{P}}^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.

Let $N_n(X)$ denote the number of degree n number fields with discriminant bounded by X. In this note, we improve the best-known upper bounds on $N_n(X)$, finding that $N_n(X)=O(X^{c{(logn)}^2})$ for an explicit constant c.

Let ${\hat{\mathcal{C}}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to ${\hat{\mathcal{C}}}_{\mathbb{Z}}$ is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order. In the appendix, we provide a careful proof that any piecewise-linear surface in a smooth 4-manifold can be isotoped to be smooth away from cone points.

We begin by establishing two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the prime number theorem, a theorem of Pillai and Selberg, a theorem of Erdős and Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. Then, by building on the ideas behind our ergodic results, we recast Sarnak’s Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak’s conjecture that focuses on the disjointness of actions of $(\mathbb{N},+)$ and $(\mathbb{N},\cdot )$. We substantiate this extension by providing proofs of several special cases.

We show that every general semiconvex entire solution to the sigma-2 equation is a quadratic polynomial. A decade ago, this result was shown for almost convex solutions. Two decades ago, this result was obtained in three dimensions, as a by-product of the work on special Lagrangian equations. Warren’s rare saddle entire solution in 2014 confirmed the necessity of the semiconvexity assumption.