Abstract:
In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichm\"uller geodesic trajectories.The main tool is quantitative nondivergence of Teichm\"uller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.

Abstract:
Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma, v\in V, and compactly supported function \phi on V, we consider the sums S_{\phi,v}(T)=\sum_{\gamma\in \Gamma_T} \phi(v \gamma). Understanding the asymptotic behavior of S_{\phi,v}(T) is a delicate problem which has only been considered for certain very special choices of H, G and {\Gamma_T}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S_{\phi,v}(T) \sim \int_{G_T} \phi(vg) dg, where G_T={g\in G:||g||

Abstract:
We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.

Abstract:
We construct generalized polygons (`parking garages') in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov-Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.

Abstract:
We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value k_mu of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality k_mu < k_nu, when mu,nu are finite and the support of mu is strictly contained in the support of nu. In combination with known results regarding the dynamics of the diagonal group we obtain isolation results as well as information regarding accumulation points of the Mordell-Gruber spectrum, extending previous work of Gruber and Ramharter. One of the main tools we develop is the associated algebra, an algebraic invariant attached to the orbit of a lattice under a block group, which can be used to characterize closed and finite volume orbits.

Abstract:
We prove that a hypothesis of Cassels, Swinnerton-Dyer, recast by Margulis as statement on the action of the diagonal group $A$ on the space of unimodular lattices, is equivalent to several assertions about minimal sets for this action. More generally, for a maximal $\mathbb{R}$-diagonalizable subgroup $A$ of a reductive group $G$ and a lattice $\Gamma$ in $G$, we give a sufficient condition for a compact $A$-minimal subset $Y$ of $G/\Gamma$ to be of a simple form, which is also necessary if $G$ is $\mathbb{R}$-split. We also show that the stabilizer of $Y$ has no nontrivial connected unipotent subgroups.

Abstract:
We prove that for any countable set $A$ of real numbers, the set of binary indefinite quadratic forms $Q$ such that the closure of $Q(\mathbb{Z}^2)$ is disjoint from $A$ has full Hausdorff dimension.

Abstract:
We give infinite lists of translations surfaces with no convex presentations. We classify the surfaces in the stratum H(2) which do not have convex presentations, as well as those with no strictly convex presentations. We show that in H(1,1), all surfaces in the eigenform loci E_4, E_9 or E_{16} have no strictly convex presentation, and that the list of surfaces with no convex presentations in H(1,1) - (E_4 union E_9 union E_{16}) is finite and consists of square-tiled surfaces. We prove the existence of non-lattice surfaces without strictly convex presentations in all of the strata H^{(hyp)}(g-1, g-1).

Abstract:
We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property.

Abstract:
For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any non-elementary Veech group can appear only finitely many times in a fixed stratum, that any non-elementary Veech group is of finite index in its normalizer, and that the quotient of the upper half plane by a non-lattice Veech group contains arbitrarily large embedded disks. These are proved using the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.