There is a long-standing conjecture in Hamiltonian analysis which claims that there exist at least $n$ geometrically distinct closed characteristics on every compact convex hypersurface in $R^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been completely solved only for $n=2$. In this article, we give a confirmed answer to this conjecture for $n=3$. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface $\Sigma$ in $R^{2n}$ when the number of geometrically distinct closed characteristics on $\Sigma$ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for $R^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in $R^4$, we prove that both of them must be irrationally elliptic

We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of a compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our results can be considered as a generalization of the simplicial coordinate of Penner [P1], [P2] for hyperbolic metrics with cusp ends to the case of surfaces with geodesic boundary. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate

Let $X$ be a symmetric space of noncompact type, and let $\Gamma$ be a lattice in the isometry group of $X$. We study the distribution of orbits of $\Gamma$ acting on the symmetric space $X$ and its geometric boundary $X(\infty )$, generalizing the main equidistribution result of Margulis's thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any $y\in X$ and $b\in X(\infty )$, we investigate the distribution of the set $\{(y\gamma ,b{\gamma }^{-1}):\gamma \in \Gamma \}$ in $X\times X(\infty )$. It is proved, in particular, that the orbits of $\Gamma$ in the Furstenberg boundary are equidistributed and that the orbits of $\Gamma$ in $X$ are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah's result [S, Corollary 1.2] based on Ratner's measure-classification theorem [R1, Theorem 1]

In the context that arose from an old problem of Lang regarding the torsion points on subvarieties of ${\mathbb{G}}_m^d$, we describe the points that lie in a given variety, are defined over the cyclotomic closure $k^c$ of a number field $k$, and map to a torsion point under a finite projection to ${\mathbb{G}}_m^d$. We apply this result to obtain a sharp and explicit version of Hilbert's irreducibility theorem over $k^c$. Concerning the arithmetic of dynamics in one variable, we obtain by related methods a complete description of the polynomials having an infinite invariant set contained in $k^c$. In particular, we answer a number of long-standing open problems posed by W. Narkiewicz and which he eventually collected explicitly in the book [N2]

In this article, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise, a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck [16]. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent.

The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions

We obtain large $n$ asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian unitary ensemble (GUE) of ($n\times n$)-matrices. Our results can also be interpreted as asymptotics of the determinant of a Hankel matrix whose symbol is supported on the real line and possesses power-like (Fisher-Hartwig) singularities