Denote by $\parallel\cdot\parallel$ the Euclidean norm in $\mathbb {R}\sp k$. We prove that the local pair correlation density of the sequence $\parallel\mathbf {m}-\mathbf {\alpha}\parallel\sp k,\mathbf {m}\in \mathbb {Z}\sp k$, is that of a Poisson process, under Diophantine conditions on the fixed vector $\mathbf {\alpha}\in \mathbb {R}\sp k$ in dimension two, vectors $\mathbf {\alpha}$ of any Diophantine type are admissible; in higher dimensions $(k>2)$, Poisson statistics are observed only for Diophantine vectors of type $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of M. Berry and M. Tabor on the Poisson nature of spectral correlations in quantized integrable systems.

We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. Makar-Limanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus $4$ handlebody which becomes overtwisted after Legendrian $-1$ surgery and study certain Legendrian surgeries on $T\sp 3$.

We deal with strongly elliptic differential operators of an arbitrary even order $2m$ with constant real coefficients and introduce a notion of the regularity of a boundary point with respect to the Dirichlet problem which is equivalent to that given by N. Wiener in the case of $m=1$. It is shown that a capacitary Wiener's type criterion is necessary and sufficient for the regularity if $n=2m$. In the case of $n>2m$, the same result is obtained for a subclass of strongly elliptic operators.

We consider fourth-order parabolic equations of gradient type. For the sake of simplicity, the analysis is carried out for the specific equation $u\sb t=-\gamma\ u\sb {xxxx}+\beta u\sb {xx}-F\sp \prime(u)$ with $(t,x)\in (0,\infty)\times(0, L)$ and $\gamma,\beta>0$ and where $F(u)$ is a bistable potential. We study its stable equilibria as a function of the ratio $\gamma/beta\sp 2$. As the ratio $\gamma/beta\sp 2$ crosses an explicit threshold value, the number of stable patterns grows to infinity as $L\to \infty$. The construction of the stable patterns is based on a variational gluing method that does not require any genericity conditions to be satisfied.

Let $T\subset \mathbf {R}\sp {m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partialT$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed initial point $A\in X$, a prescribed final point $B\in X$, and make a prescribed number $n$ of reflections at the boundary $X$. We apply a topological approach based on the calculation of cohomology rings of certain configuration spaces of $S\sp m$.

In this paper we give topological lower bounds on the number of periodic and of closed trajectories in strictly convex smooth billiards $T\subset \mathbf {R}\sp {m+1}$. Namely, for given $n$, we estimate the number of $n$-periodic billiard trajectories in $T$ and also estimate the number of billiard trajectories which start and end at a given point $A\in \partial T$ and make a prescribed number n of reflections at the boundary $\partial T$ of the billiard domain. We use variational reduction, admitting a finite group of symmetries, and apply a topological approach based on equivariant Morse and Lusternik-Schnirelman theories.