Directed cell motility is a process whereby the motility machinery of the cell (involving the interaction of actin with myosin) is organized spatially so as to cause directed motion. In Dictyostelium, this occurs as the cell responds to cAMP gradients during the aggregation process. In keratocytes, the cell spontaneously polarizes itself (without external cues). This talk will focus on spatially extended modeling of both the signaling system which encodes the directional information and the downstream mechanical response and the comparison of these models to detailed experimental studies of both of these systems.

What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: for arbitrarily large numbers N, can there be sets of N positive integers where both the number of pairwise sums and pairwise product is less than N^{3/2}?

No one knows. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 more than a multiple of 5, a set with density 2/5. Can you do better?

The shadowing problem is related to the following question: under which condition, for any pseudotrajectory approximate trajectory) of a vector field there exists a close trajectory? It is known that in a neighbourhood of a hyperbolic set diffeomorphisms and vector fields have shadowing property. In fact more general statement is correct: structurally stable dynamical systems satisfy shadowing property.

We are interested if converse implication is correct. We consider notion of Lipschitz shadowing property and proved that it is equivalent to structural stability for the cases of diffeomorphisms and vector fields.

Talk is based on joint works with S. Pilyugin and K. Palmer

In this talk, an efficient rearrangement algorithm is introduced to the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. The method proposed is based on Rayleigh quotient formulation of eigenvalues and rearrangement algorithms which can handle topology changes automatically. Using the efficient rearrangement strategy, the new proposed method is more efficient than classical level set approaches based on shape and/or topological derivatives. The optimal results are explored theoretically and numerically under different geometries and boundary conditions.

Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.