We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in $\Bbb{Z}/2\mathbb{Z})$. The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod $2$ Ozsváth-Szabó mixed invariants of closed four-manifolds, also in terms of grid diagrams.

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\ge 2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.

The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied for more than 90 years. In this paper, we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.

It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example $M^7$ with $\mathrm{Ric} \ge 0$ such that $\pi_1(M)=\mathbb{Q}/\mathbb{Z}$ is infinitely generated.

There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group $\pi_0\mathrm{Diff}(S^3\times S^3)$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of $\pi_0\mathrm{Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with $\mathrm{Ric}>0$.

We prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt(3)$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt{3}$ must converge, up to isometry, to the triangular Moebius band.These results answer the mimimum aspect ratioquestion discussed by W. Wunderlich in 1962 and the more specific conjecture of B. Halpern and C. Weaver from 1977.

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\xi$ can be approximated by algebraic numbers $\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(\alpha)$ of $\alpha$. He showed that the supremum $\omega_n^\ast(\xi)$ of all $\omega$ for which infinitely many such $\alpha$ have $\vert\xi-\alpha|\le H(\alpha)^{-\omega-1}$ is at least $(n+1)/2$. He also asked if we could even have $\omega_n^\ast(\xi)\ge n$ as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form $n/2+\mathcal{O}(1)$ until Badziahin and Schleischitz showed in 2021 that $\omega_n^\ast(\xi)\ge an$ for each $n\ge 4$, with $a=1/\sqrt{3} \simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $\omega_n^\ast(\xi)\ge an$ for each $n\ge 2$, with $a=1/(2-\mathrm{log} 2) \simeq 0.765$.