Abstract: In this blackboard talk, I will discuss new results in the field
of directed polymers (these are topological objects found in many
condensed matter systems, e.g. flux lines in superconductors, domain
walls in ferromagnets, and atomic steps on vicinal surfaces). After a
detailed introduction, I will describe the physics of "self-localization"
of directed polymers, a state induced by elastic interactions between the
polymer and its environment. In the latter part of the talk, I will show
how the model of self-localization may be mapped to population dynamics,
in which, along with local birth and death rules, one imposes a global
constraint which fixes the total population. The dynamics of populations
under such oppressive control turns out to be very rich, with phases such
as "spreading", "collapse", and a novel "pseudo-traveling wave".