The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement M\setminus L of the Lagrange spectrum L in the Markov spectrum M remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that M\setminus L contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of M\setminus L were < \sqrt{12} = 3.46\ldots , and this was the status quo of our knowledge of M\setminus L until 2017. In this article, we show the following two results. First, we prove that M\setminus L is richer than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than 1/2 near 3.7: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that M\setminus L is not very thick: its Hausdorff dimension is strictly smaller than one.

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by carefully choosing a small deformation of the coefficients of the original equations. This deformation preserves all polynomial relations over \mathbb Q satisfied by these coefficients and is equisingular in the sense of Zariski. Moreover we construct an algorithm, that, given a system of equations defining a variety V , produces a system of equations with coefficients in \bar{\mathbb Q} of a variety homeomorphic to V .

Let \pi be a cuspidal automorphic representation of PGL _2(\mathbb A_\mathbb Q) of arithmetic conductor C and archimedean parameter T , and let \phi be an L^2 -normalized automorphic form in the space of \pi . The sup-norm problem asks for bounds on \| \phi \|_\infty in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L^2 -mass |\phi|^2 (g) \, d g of \phi . Previous work on these problems in the conductor-aspect has focused on the case that \phi is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C , the local component \pi_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms \phi for which the local components \phi_p \in \pi_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL _2(\mathbb{R}) . For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if \pi_\infty is a holomorphic discrete series of lowest weight k , we obtain the optimal bound C^{1/8 -\epsilon} k^{1/4 - \epsilon} \ll_{\epsilon} |\phi|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4+\epsilon} . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

We provide explicit counterexamples to the so-called Complement Problem in every dimension n\geq3 , i.e. pairs of nonisomorphic irreducible algebraic hypersurfaces H_1, H_2\subset\mathbb C^{n} whose complements \mathbb C^{n}\setminus H_1 and \mathbb C^{n}\setminus H_2 are isomorphic. Since we can arrange that one of the hypersurfaces is singular whereas the other is smooth, we also have counterexamples in the analytic setting.

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary which is not S^1 \times D^2 . Consider any element \phi in the first cohomology of M with integer coefficients. Then one can define the \phi -twisted L^2 -torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that the degree of the L^2 -torsion function, which is a number extracted from its asymptotic behavior at 0 and at \infty , agrees with the Thurston norm of \phi .

We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.

The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen’s version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class . In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only \mathbb R -covered Anosov flows up to orbit equivalence and finite lifts that do not admit at least one infinite free homotopy class of periodic orbits.

We prove vanishing of cohomology with coefficients in representations on a large class of Banach spaces for a group acting “nicely” on a simplicial complexes based on spectral properties of the 1-dimensional links of the simplicial complex.

We study some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of “unlikely intersections”. We prove a special case of the Zilber–Pink conjecture for curves.

In this paper, we consider a Diophantine quasi-periodic time-dependent analytic perturbation of a convex integrable Hamiltonian system, and we prove a result of stability of the action variables for an exponentially long interval of time. This extends known results for periodic time-dependent perturbations, and partly solves a long standing conjecture of Chirikov and Lochak. We also obtain improved stability estimates close to resonances or far away from resonances, and a more general result without any Diophantine condition.