We prove analogs of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n,R)/ SL(n,Z). The key lemma for our results says the measure of the set of unimodular lattices in R n that does not intersect a 'large' volume subset of R n is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.

We prove quantitative recurrence and large deviations results for the Teichmuller geodesic flow on connected components of strata of the moduli space Q g of holomorphic unit-area quadratic differentials on a compact genus g ≥ 2 surface.

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis [Mar70] to Teichmüller space. Let X be a point in Teichmüller space, and let BR(X) be the ball of radius R centered at X (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as R tends to infinity for the volume of BR(X), and also for the cardinality of the intersection of BR(X) with an orbit of the mapping class group.

Abstract. We construct a Poincaré section for the horocycle flow on the modular surface SL(2, R)/SL(2, Z), and study the associated first return map, which coincides with a transformation (the BCZ map) defined by Boca-Cobeli-Zaharescu [8]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation and probability theory.