We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least $\mathrm{log}^C n$ for a large enough constant $C$. Our extractor outputs one bit and has error $n^{-\Omega(1)}$. The best previous extractor, by Bourgain, required each source to have min-entropy $.499n$. M

A key ingredient in our construction is an explicit construction of a monotone, almost-balanced Boolean function on $n$ bits that is resilient to coalitions of size $n^{1-\delta}$ for any $\delta>0$. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown $n-q$ bits are chosen almost $\mathrm{polylog}(n)$-wise independently, and the remaining $q=n^{1-\delta}$ bits are chosen by an adversary as an arbitrary function of the $n-q$ bits. The best previous construction, by Viola, achieved $q=n^{1/2-\delta}$.

Our explicit two-source extractor directly implies an explicit construction of a $2^{(\mathrm{log}\ \mathrm{log}\ N)^{O(1)}}$-Ramsey graph over $N$ vertices, improving bounds obtained by Barak et al. and matching an independent work by Cohen.

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p\ \mathrm{mod}\ 1$ and $\times_q\ \mathrm{mod}\ 1$, respectively, and if $\mathrm{log}\ p/\mathrm{log}\ q\ne \mathbb{Q}$, then for all real numbers $u$ and $v$,$$\mathrm{dim}_{\mathrm{H}}(uA+v) \cap B \le \mathrm{max}\{0,\mathrm{dim}_{\mathrm{H}} A + \mathrm{dim}_{\mathrm{H}} B -1\}.$$.We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on $\mathbb{R}$. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that an absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply ``quantum ergodicity," a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies, in particular, to graphs converging to the Anderson model on a regular tree, in the regime of extended states studied by Klein and Aizenman-Warzel.

We show that, for $n\ge 3$, $\mathrm{lim}_{t\to 0} e^{it\Delta} f(x) f(x)$ holds almost everywhere for all $f\in H^s(\mathbb{R}^n)$ provided that $s> \frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of the divergence set of the Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in a method to compute a finite set of $p$-adic points, containing the rational points, on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell--Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This is then applied to determine the rational points of the modular curve $X_s(13)$, completing the classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan level structure due to Bilu--Parent and Bilu--Parent--Rebolledo.

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields.

Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varieties yielding intersections of excessive dimension in natural algebraic families.

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle; i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda \in (0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\mathrm{dim}\ \nu_\lambda = 1$ for all transcendental $\lambda = (1/2,1)$.

We show that a closed, connected and orientable Riemannian manifold $M$ of dimension $d$ that admits a nonconstant quasiregular mapping from $\mathrm{R}^d$ must have bounded dimension of the cohomology independent of the distortion of the map. The dimension of the degree $1$ de Rham cohomology of $M$ is bounded above by $\binom{d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a $d$-dimensional, simply connected manifold that does not admit a quasiregular mapping from $\mathbb{R}^d$. Our result gives an affirmative answer to this question.