We prove that the Aubry and Mañé sets introduced by Mather in Lagrangian dynamics are symplectic invariants. In order to do so, we introduce a barrier on phase space. This is also an occasion to suggest an Aubry-Mather theory for nonconvex Hamiltonians

Résumé

On montre que les ensembles d'Aubry et de Mañé introduits par Mather en dynamique Lagrangienne sont des invariants symplectiques. On introduit pour ceci une barriere dans l'espace des phases. Ceci est aussi l'occasion d'ébaucher une théorie d'Aubry-Mather pour des Hamiltoniens non convexes

In this article, we study a class of Monge-Ampère equations of mixed type which is elliptic in one side of a hypersurface and hyperbolic in another side. The degeneracy along the hypersurface is allowed to be at arbitrary, even infinite, order. The linearized equation is also of mixed type and includes the high-dimensional Tricomi equation as a special case. We establish the existence of sufficiently smooth local solutions to the Monge-Ampère equations via Nash-Moser iteration

Let $R_1,R_2,\dots ,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$, $d\ge 2$, and let $f_1,f_2,\dots ,f_m$ be small smooth perturbations of them. We show that $\{f_{\alpha }\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory

We compute the reduced version of Khovanov and Rozansky's $\mathfrak{sl}$(N) homology for two-bridge knots and links. The answer is expressed in terms of the skein polynomial of Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter (or HOMFLY polynomial; see [6]) and signature

We prove that the moduli space ${\stackrel{̲}{M}}_{0,0}(N,2)$ of stable maps of degree $2$ to the moduli space $N$ of rank $2$ stable bundles of fixed odd determinant $O_X(-x)$ over a smooth projective curve $X$ of genus $g\ge 3$ has two irreducible components that intersect transversely. One of them is Kirwan's partial desingularization ${\tilde{M}}_X$ of the moduli space $M_X$ of rank $2$ semistable bundles with determinant isomorphic to $O_X(y-x)$ for some $y\in X$. The other component is the partial desingularization of the geometric invariant theory (GIT) quotient $\mathbb{P}\mathrm{Hom}(\mathrm{sl}(2){}^{\vee },W)//\mathrm{PGL}(2)$ for a vector bundle $W=R^1{\pi }_{*}{L}^{-2}(-x)$ of rank $g$ over the Jacobian of $X$. We also show that the Hilbert scheme $H$, the Chow scheme $C$ of conics in $N$, and ${\stackrel{̲}{M}}_{0,0}(N,2)$ are related by explicit contractions