A convolution of two singular continuous measures can be singular continuous or absolutely continuous (or of a mixed type). It is usually hard to determine which case is present for a specific pair of measures. It turnes out that for measures of maximal entropy of large Hausdorff dimension supported on dynamically defined Cantor sets generically the convolution is a.c. (this is a joint result with D.Damanik and B.Solomyak). This is in a sense a measure-theoretical counterpart of the claim (known as Newhouse Gap Lemma) that the sum of two sufficiently thick Cantor sets must contain an interval. 

The fundamental processes of DNA synthesis, mitosis and cell division in eukaryotic cells are controlled by a complex network of interacting genes and proteins focused on periodic activation of a family of master regulators, the cyclin-dependent protein kinases (CDKs). This regulatory network must ensure the strict alternation of DNA synthesis (S phase) and mitosis (M phase) and the proper coordination of cell division with cell growth. Moreover, the control system must operate robustly in the face of considerable molecular noise that is unavoidable in the small confines of a yeast cell. After reviewing the basic molecular biology of the CDK regulatory network in budding yeast, I will examine the physiological consequences of the reaction mechanism by mathematical and computational modeling. Deterministic models (differential equations) describe the average behavior of populations of yeast cells, and stochastic models address issues of variability and robustness in individual cells. Both types of models will be evaluated in light of quantitative experimental observations.

Traveling waves in actin have recently been reported in many cell types. Fish keratocyte cells, which usually exhibit rapid and steady motility, exhibit traveling waves of protrusion when plated on highly adhesive surfaces. We hypothesize that waving arises from a competition between actin polymerization and mature adhesions for VASP, a protein that associates with growing actin barbed ends. We developed a mathematical model of actin protrusion coupled with membrane tension, adhesions and VASP. The model is formulated as a system of partial differential equations with a nonlocal integral term and noise. Simulations of this model lead to a number of predictions, for example, that overexpression of VASP prevents waving, but depletion of VASP does not increase the fraction of cells that wave. The model also predicts that VASP exhibits a traveling wave whose peak is out of phase with the F-actin wave. Further experiments confirmed these predictions and provided quantitative data to estimate the model parameters. We thus conclude that the waves are the result of competition between actin and adhesions for VASP, rather than a regulatory biochemical oscillator or mechanical tag-of-war. We hypothesize that this waving behavior contributes to adaptation of cell motility mechanisms in perturbed environments.