We prove that given a Hitchin representation in a split real rank 2 group $\mathsf{G}_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrisation of the Hitchin component by a Hermitian bundle over Teichmüller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of $\mathsf{G}_0$. Some partial extensions of the construction hold for cyclic bundles in higher rank.

There is a rich theory of so-called (strict) nearly Kähler manifolds, almost-Hermitian manifolds generalising the famous almost complex structureon the 6-sphere induced by octonionic multiplication. Namely Kähler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group thecompact exceptional Lie group $\mathrm{G}_2$: the metric cone over aRiemannian 6-manifold M has holonomy contained in $\mathrm{G}_2$ if and onlyif $M$ is a nearly Kähler 6-manifold.

A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existenceof at least one cohomogeneity one nearly Kähler structure on the6-sphere and on the product of a pair of 3-spheres. We conjecture thatthese are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions.

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M \to N$ between Riemannian manifolds. If $S^k(f)\equiv \{x \in M \colon \text{no tangent map at } x \text{ is } k+1\text{-symmetric}\}$ is the $k^{\mathrm{th}}$-stratum of the singular set of $f$, then it is well known that $\mathrm{dim}\ S^k \le k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^l(f)$ that there exists a unique $k$-plane $V^k \subseteq T_xM$ such that every tangent map at $x$ is $k$-symmetric with respect to $V$.

In the case of minimizing harmonic maps we go further and prove that the singular set $S(f)$, which is well known to satisfy $\mathrm{dim}\ S(f)\le n-3$, is in fact $n-3$-rectifiable with uniformly finite $n-3$-measure. An effective version of this allows us to prove that $|\nabla f|$ has estimates in $L^3_{\mathrm{weak}}$, an estimate that is sharp as $|\nabla f|$ may not live in $L^3$. More generally, we show that the regularity scale $r_f$ also has $L^3_{\mathrm{weak}}$ estimates.

The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S^k_{\varepsilon,r}(f)$ and $S^k_\varepsilon(f) \equiv S^k_{\varepsilon,0}(f)$. Roughly, $S^k_\varepsilon \subseteq M$ is the collection of points $x \in M$ for which no ball $B_r(x)$ is $\varepsilon$-close to being $k+1$-symmetric. We show that $S^k_\varepsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $\mathrm{Vol}(B_r S_\varepsilon^k)\le C r^{n-k}$.

The proofs require a new $L^2$-subspace approximation theorem for stationary harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the classical Reifenberg and give checkable criteria to determine when a set is $k$-rectifiable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest. The $L^2$-subspace approximation theorem we prove is then used to help break down the quantitative stratifications into pieces that satisfy these criteria.

We prove that faithful traces on separable and nuclear $\mathrm{C}^*$-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear $\mathrm{C}^*$-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal $\mathrm{C}^*$-algebras.

We consider a large class of 2D area-preserving diffeomorphisms that arenot uniformly hyperbolic but have strong hyperbolicity properties on large regions oftheir phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potentialswitching of ``stable" and ``unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains withrelative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

We give a sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates. We show that there are initial data in $H^2$ satisfying the wave coordinate condition such that there is no solution in $H^2$ to Einstein equations in wave coordinates for any positive time. This result is sharp since Klainerman-Rodnianski and Smith-Tataru proved existence for the same equations with slightly more regular initial data.

We show that for an integer $n \ge 1$, any subset A $\subseteq \mathbb{Z}_4^n$ free ofthree-term arithmetic progressions has size $|A| \le 4^{\gamma n}$, with anabsolute constant $\gamma\approx 0.926$.

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression by $c^n$ with $c \lt q$. For $q=3$, the problem of finding the largest subset of $\mathbb{F}_3^n$ with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order $n^{-1-\varepsilon} 3^n$.