The group of rational points is an important but subtle invariant of an abelian variety defined over a number field.  In the case of an elliptic curve over Q, a celebrated theorem of Mazur asserts that there are only finitely many possibilities for the torsion part; the same is conjectured to be true for all abelian varieties over number fields though very little has been proven in higher dimensions.  The natural geometric analog, known as the geometric torsion conjecture, asks for a bound on the torsion sections of a family of abelian varieties over a complex curve, and can be interpreted as the nonexistence of low genus curves in congruence towers of Siegel modular varieties.  We will discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry and apply it to some special cases of the torsion conjecture as well as some related problems.  Along the way we will also deduce some results about the global geometry of these moduli spaces.  This is joint work with J. Tsimerman.

In this talk we show that if the normalized cocycle related to a quasiperiodic Jacobi operator H_{v,c} (where v being the potential and c being the off-diagonal term) is reducible to a constant rotation for almost all energies with respect to the density of states measure, then its dual model has either purely point spectrum for almost all phase or purely absolutely continuous spectrum for almost all phase, depending on the winding number of c. As a corollary, we obtain the complete phase-transition of extended Harper's model in the positive Lyapunov exponent region.

Gene regulatory networks controlling cellular differentiation must sense changes in environmental conditions and coordinate gene expression with cell-division cycles. However, the mechanisms of sensing and integrating different signals remain elusive even for the best studied model systems. Here we uncover a simple solution to this complicated task by investigating the Bacillus subtilis sporulation network. We show that this network evaluates the level of starvation without specific metabolite sensing by detecting changes in cellular growth rate. In addition, the arrangement of sporulation network genes on the chromosome allows the network to coordinate its response with cell cycle by exploiting the transient gene dosage imbalance during chromosome replication. These design features allows cells to decide between sporulation and continued vegetative growth during each cell cycle spent in starvation conditions. The simplicity of this cell-fate decision mechanism suggests that it may be widely applicable in a variety of gene regulatory and stress-response settings.

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We give some properties of the \kappa-topologies over 2^{\lambda} introduced earlier (for \kappa \leq \lamba), define a Cantor-Bendixon process for forests, and begin work on showing the consistency of Cantor-Bendixon theorem analogues for closed subsets of 2^{\kappa} and P_{\kappa^+}\lambda, for \kappa regular.

The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kesten’s criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel.