In 2001, Donaldson proved that the existence of cscK metrics on a
polarized manifold (X,L) with discrete automorphism group implies the
existence of balanced metrics on L^k for k large enough. We show that the
similar statement holds if one twists the line bundle L with a simple
stable vector bundle E. More precisely we show that if E is a simple
stable bundle over a polarized manifold (X,L), (X,L) admits cscK metric
and have discrete automorphism group, then (PE^*, \O(d) \otimes L^k)
admits a balanced metric for k large enough.

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