Nous décrivons le complexe de de Rham des revĉtements du demi-plan de Drinfeld pour $\mathrm{GL}_2(\mathbf{Q}_p)$. Cette description, conjecturée par Breuil et Strauch, fournit une réalisation géeométrique de la correspondance de Langlands locale $p$-adique pour certaines repréesentations de de Rham de dimension $2$ de $\mathrm{Gal}(\overline{\mathbf{Q}_p}/\mathbf{Q}_p)$.

We describe the de Rham complex of the éetale coverings of Drinfeld's $p$-adic upper half-plane for $\mathrm{GL}_2(\mathbf{Q}_p)$. Conjectured by Breuil and Strauch, this description gives a geometric realization of the $p$-adic local Langlands correspondence for certain two-dimensional de Rham representations of $\mathrm{Gal}\overline{\mathbf{Q}_p}/\mathbf{Q}_p)$.

We prove nontrivial bounds for general bilinear forms inhyper-Kloosterman sums when the sizes of both variables may be belowthe range controlled by Fourier-analytic methods (Pólya-Vinogradovrange). We then derive applications to the second moment of cuspforms twisted by characters modulo primes, and to the distributionin arithmetic progressions to large moduli of certainEisenstein-Hecke coefficients on $\mathrm{GL}_3$. Our main tools are newbounds for certain complete sums in three variables over finitefields, proved using methods from algebraic geometry, especially$\ell$-adic cohomology and the Riemann Hypothesis.

We prove that the $2$-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined $4$-fold Toda bracket $(2,\theta_4\theta_4,2).

Our result has a geometric corollary: the $61$-sphere has a unique smooth structure, and it is the last odd dimensional case --- the only ones are $S^1$, $S^3$, $S^5$ and $S^{61}$.

Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k\ge 1$ and $A,B\subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.

We show that $\mathrm{lim}_{t\to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s(\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the $i^{\mathrm{th}}$ Betti number $\beta_i(M)$ of a nonzero module $M$ of finite length and finite projective dimension over a local ring $R$ of dimension $d$ should be at least ${d \choose i}$. It would follow from the validity of this conjecture that $\sum_i \beta_i(M) \ge 2^d$. We prove the latter inequality holds in a large number of cases and that, when $R$ is a complete intersection in which $2$ is invertible, equality holds if and only if $M$ is isomorphic to the quotient of $R$ by a regular sequence of elements.

We explain a mistake that occurred in the proof of Murthy's conjecture by the author.