We prove a conjecture of Calegari's, that every quasigeodesic flow on a closed hyperbolic 3-manifold contains a closed orbit.

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\ldots,x\}$. Motivated by applications to factorization algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/\pi + o(1)$ as $x\to \infty$ and made a conjecture regarding the location of the sharp threshold.

In this paper we prove both of these conjectures by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.

We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and we may assume any real value in the interval $[0, 1]$.

Given $k\in \mathbb{N}$, the $k$'th discrete Heisenberg group, denoted $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$, is the group generated by the elements $a_1,b_1,\ldots,a_k,b_k,c$, subject to the commutator relations $[a_1,b_1]=\cdots=[a_k,b_k]=c$, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct $i,j\in \{1,\dots,k\}$, we have $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$. (In particular, this implies that $c$ is in the center of $\mathbb{H}_{ \scriptscriptstyle{\mathbb{Z}}}^{2k+1}$.) Denote $\mathfrak{S}_k=\{a_1,b_1,a_1^{-1},b_1^{-1},\ldots,a_k,b_k,a_k^{-1},b_k^{-1}\}$. The horizontal boundary of $\Omega\subseteq \mathbb{H}_{ \scriptscriptstyle{ \mathbb{Z}}}^{2k+1}$, denoted $\partial_{\mathsf{h}}\Omega$, is the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \mathfrak{S}_k$. The horizontal perimeter of $\Omega$ is the cardinality $|\partial_{\mathsf{h}}\Omega|$ of $\partial_{\mathsf{h}}\Omega$; i.e., it is the total number of edges incident to $\Omega$ in the Cayley graph induced by $\mathfrak{S}_k$. For $t\in \mathbb{N}$, define $\partial^t_{\mathsf{v}} \Omega$ to be the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \{c^t,c^{-t}\}$. Thus, $|\partial^t_{\mathsf{v}}\Omega|$ is the total number of edges incident to $\Omega$ in the (disconnected) Cayley graph induced by $\{c^t,c^{-t}\}\subseteq \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$. The vertical perimeter of $\Omega$ is defined by $|\partial_{\mathsf{v}}\Omega|= \sqrt{\sum_{t=1}^\infty |\partial^t_{\mathsf{v}}\Omega|^2/t^2}$. It is shown here that if $k\ge 2$, then $|\partial_{\mathsf{v}}\Omega|\lesssim \frac{1}{k} |\partial_{\mathsf{h}}\Omega|$. The proof of this ``vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an ``intrinsic corona decomposition.'' This allows one to deduce an endpoint $W^{1,1}\to L_2(L_1)$ boundedness of a certain singular integral operator from a corresponding lower-dimensional $W^{1,2}\to L_2(L_2)$ boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every $n\in \mathbb{N}$, any embedding into an $L_1(\mu)$ space of a ball of radius $n$ in the word metric on $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{5}$ that is induced by the generating set $\mathfrak{S}_2$ incurs bi-Lipschitz distortion that is at least a universal constant multiple of $\sqrt{\log n}$. As an application to approximation algorithms, it follows that for every $n\in \mathbb{N}$, the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a universal constant multiple of $\sqrt{\log n}$.M

We compute the algebraic hull of the Kontsevich--Zorich cocycle over any $\mathrm{GL}^+_2(\mathbb{R})$ invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.