Abstract. We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.

We classify the measures on SL(k, R)/ SL(k, Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.

Abstract. We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions.

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL3(Z)\SL3(R)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(Z)\SL3(R)/SO3 of volume ≤ V becomes equidistributed as V → ∞.The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

Abstract. We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who in particular proved this equidistribution under an additional congruence assumption on the discriminant. We give a more conceptual treatment using entropy theory, and show how to use positivity of the discriminant as a substitute for Linnik's congruence condition.