The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems.

We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum.

The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a nontrivial contractible one-periodic orbit when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.

We give a positive response to a question of W. Veech: Infinitely generated Veech groups do exist.

In this paper we show that the period equals the index for elements of Brauer groups of (function fields of) surfaces. A key idea of the proof is that any Azumaya algebra over a surface can be transformed into an Azumaya algebra that is unobstructed.

Cut the unit circle $S^1=\mathbb{R}/\mathbb{Z}$ at the points $\{\sqrt{1}\}, \{\sqrt{2}\},\ldots,\{\sqrt{N}\}$, where $\{x\} = x \bmod 1$, and let $J_1, \ldots, J_N$ denote the complementary intervals, or \emph{gaps}, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths $|J_i|/N$ are governed by an explicit piecewise real-analytic distribution $F(t) \,dt$ with phase transitions at $t=1/2$ and $t=2$.

The gap distribution is related to the probability $p(t)$ that a random unimodular lattice translate $\Lambda \subset \mathbb{R}^2$ meets a fixed triangle $S_t$ of area $t$; in fact, $p''(t) = -F(t)$. The proof uses ergodic theory on the universal elliptic curve

\[ E = \big(\mathrm{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/ \big(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big) \]

and Ratner's theorem on unipotent invariant measures.

The cohomology $H ^* (\Gamma, E)$ of an arithmetic subgroup $\Gamma$ of a connected reductive algebraic group $G$ defined over some algebraic number field $F$ can be interpreted in terms of the automorphic spectrum of $\Gamma$. With this framework in place, there is a sum decomposition of the cohomology into the cuspidal cohomology (i.e., classes represented by cuspidal automorphic forms for $G$) and the so-called Eisenstein cohomology constructed as the span of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components of proper parabolic $F$-subgroups of $G$. The main objective of this paper is to isolate a specific structural part in the Eisenstein cohomology. This pertains to regular Eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered. It is shown that the cohomological degree of these classes is bounded from below by the constant $q_0(G(\mathbb{R}))=((1/2) [\dim X_{G(\mathbb{R})}-(\rk (G(\mathbb{R}))-\rk(K_{\mathbb{R}}))]$, where $K_{\mathbb{R}}$ denotes a maximal compact subgroup of the real Lie group $G(\mathbb{R})$, where $X_{G(\mathbb{R})}$ is the associated symmetric space. This investigation has various applications. One of these is a vanishing result for the cohomology in the generic case (i.e., where the representation determining the coefficient system $E$ has regular highest weight) in degrees below $q_0(G(\mathbb{R}))$. This is a sharp bound depending only on the underlying real Lie group $G(\mathbb{R})$ (Corollary 5.6, Proposition 5.8). This result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular Eisenstein cohomology classes.

To each partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ with distinct parts we assign the probability $Q_\lambda(x) P_\lambda(y)/Z$, where $Q_\lambda$ and $P_\lambda$ are the Schur $Q$-functions and $Z$ is a normalization constant. This measure, which we call the shifted Schur measure, is analogous to the much-studied Schur measure. For the specialization of the first $m$ coordinates of $x$ and the first $n$ coordinates of $y$ equal to $\alpha$ ($0<\alpha<1$) and the rest equal to zero, we derive a limit law for $\lambda_1$ as $m,n \to \infty$ with $\tau=m/n$ fixed. For the Schur measure, the $\alpha$-specialization limit law was derived by Johansson [J1]. Our main result implies that the two limit laws are identical.

We use analytic methods to define family Gromov-Witten invariants for Kähler surfaces. We prove that these are well-defined invariants of the deformation class of the Kähler structure.