This talk introduces several results relating the dominating sets in a graph to the independent sets.

A {\em jointly independent set} for graphs $G_1, G_2, \ldots,G_r$ is an independent set in the union of the graphs $G_i$, namely a set independent in all of the graphs. This notion has a fractional counterpart. For a graph $G$ we denote by $\Omega(G)$ the polytope in $\mathbb{R}^V$ whose vertices are the incidence vectors of the independent sets of $G$. Let $\Omega(G_1,...,G_r)= \bigcap_{i \le m} \Omega(G_i)$. Write $\alpha^*(G_1,...,G_r)= \max \{\vec{1} \cdot \vec{x}: ~ \vec{x} \in \Omega(G_1,...,G_r)\}$. Also write: $\gamma(G_1,...,G_r)$ for the minimal number of non-punctured neighborhoods in the graphs, whose union is $V$. One of the results that will be shown is that for any pair $\{G_1,G_2\}$ of graphs $\alpha^*(\{G_1, G_2\}) \ge \gamma(\{G_1, G_2\})$. For $r$ graphs $G_1,G_2,\ldots,G_r$ we have $\alpha^*(\{G_1, G_2,\ldots,G_r\}) \ge \frac{2}{r}\gamma(\{G_1, G_2,\ldots,G_r\})$. This generalizes a result of Lovasz.

This is joint work with Ron Aharoni, Ron Holzmann and Ori Kfir.

http://www.math.princeton.edu/%7Esnorin/berger2007-fall.pdf

Recent and beautiful theorem of Ben Green and Terence Tao asserts that the primes contain arithmetic progressions of any length. The proof has two parts.
In the first part Green and Tao obtain an extension of well-known Szemeredi's theorem on arithmetic progressions. Their proof of the extension is in spirit of ergodic theory. In the second part, using analytical methods, they reduce a question of existence of progressions in the primes to their extension of Szemeredi's theorem.
In our talk we shall discuss the ergodic part of Green-Tao proof.

Animal groups such as bird flocks, insect swarms and fish schools are spectacular, ecologically important and sometimes devastating features of the biology of various species. Outbreaks of the desert locust, for example, can invade approximately one fifth of the Earth's land surface and are estimated to affect the livelihood of one in ten people on the planet. Using a combined theoretical and experimental approach involving insect and vertebrate groups I will address how, and why, individuals move in unison and investigate the principals of information transfer in these groups, particularly focusing on leadership and collective consensus decision-making.

For very large animal groups, despite huge differences in the size and cognitive abilities of group members, recent models from theoretical physics ('self-propelled particle', SPP, models) have suggested that general principles underlie collective motion. Such models demonstrate that some group-level properties may be largely independent of the types of animals involved. I shall present recent experimental work on locusts that validates some of the predictions of simple mechanistic models including a density-dependent "phase transition" from disordered to ordered motion.

Details of the mechanism by which individuals interact, however, also provide important biological insights into swarm behaviour. Using laboratory studies involving nerve manipulation and field experiments we demonstrate that some swarming insects are in effect on a "forced march" driven by cannibalism.

These results will be discussed in the context of the evolution of functional complexity and pattern formation in biological systems.