We prove that the generic part of the mod $\ell$ cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge–Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $\alpha$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{\alpha,g} : \mathbb{T}^2\rightarrow\mathbb{T}^2$ be defined by $(x,y) \mapsto (x+\alpha,y+g(x))$. We prove that if $T_{\alpha,g}$ is uniquely ergodic then, for every $(x,y) \in \mathbb{T}^2$, the sequence $\{T_{\alpha,g}^p(x,y)\}$ is equidistributed on $\mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $\mathbb{T}$.

We prove that for $n\in \{4,5\}$, a closed aspherical $n$-manifold does not admit a Riemannian metric with positive scalar curvature.

Additionally, we show that for $n\le 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with contributions by Lesourd--Unger--Yau, this proves that the Schoen--Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature.

A key geometric tool in these results are generalized soap bubbles---surfaces that are stationary for prescribed-mean-curvature functionals (also called $\mu$-bubbles).

We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson's theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation corresponding to the system.

We prove an effective form of Wilkie's conjecture in the structure generated by restricted sub-Pfaffian functions: the number of rational points of height $H$ lying in the transcendental part of such a set grows no faster than some power of $\mathrm{log}\, H$. Our bounds depend only on the Pfaffian complexity of the sets involved. As a corollary we deduce Wilkie's original conjecture for $\mathbb{R}_{\mathrm{exp}}$ in full generality.

Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $\mathrm{Mod}_{g,n}$ of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if \[\rho: \pi_1(\Sigma_{g,n})\to \mathrm{GL}_r(\mathbb{C})\]is a representation whose conjugacy class has finite orbit under $\mathrm{Mod}_{g,n}$, and $r \lt\sqrt{g+1}$, then $\rho$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^n (n>1)$ is Oka. Furthermore, we obtain new examples of non-elliptic Oka manifolds which negatively answer Gromov's question. The relative version of the main theorem is also proved. As an application, we show that the complement $\mathbb{C}^n\backslash \mathbb{R}^k$ of a totally real affine subspace is Oka if $n > 1$ and $(n,k) \ne (2,1),(2,2),(3,3)$.

For integers $s,t \ge 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. In this paper we prove \[r(4,t) = \Omega\Bigl(\frac{t^3}{\mathrm{log}^4 t}\Bigr)$ \quad\quad\quad \mathrm{as}\ t \rightarrow \infty,\]which determines $r(4,t)$ up to a factor of order $\mathrm{log}^2 t$, and solves a conjecture of Erdős.