Consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrodinger operators. [This is joint work with Pedro Duarte.]

We present a theorem of Foreman that allows an exact characterization of what happens to the structure of precipitous ideals after suitable forcing. This theorem unifies several well-known results, giving as them quick corollaries. We will use it to show: forcing precipitous ideals from large cardinals, preservation theorems of Kakuda and Baumgartner-Taylor, and Solovay's consistency result on real-valued measurable cardinals.  We will also show some new applications due to the speaker.
 

My talk will be devoted to a (quick and very brief) introduction to the domino tilings (intensively studied during the last fifty years), the subject that is very simple in the origin, while giving almost immediately very beautiful images. My goal will be to explain (roughly), where does the "arctic circle" effect in tilings come from, meanwhile mentioning asymptotic shape of Young diagrams, entropy, height function and variational problems. If the time permits, I will speak about computation of determinants and permanents.