Diffusion, along with transport and reaction effects, is the main factor explaining changes
or transitions in a wide array of situations such as flames, some phase transitions, tumours
or other biological invasions. In these systems, two or several possible states coexist, and
one observes certain states expanding or receding or patterns being formed.

This lecture, meant for a general audience, will describe some mathematical properties of
reaction-diffusion equations as an approach to spatial propagation and diffusion. After
describing the mechanism of reaction and diffusion and giving several illustrations, I will
review some classical results. In the context of ecology of populations, I will then mention
some recent works dealing with non homogeneous media. In this framework, I will describe a
model addressing the question of how a species keeps pace with a shifting climate.

The interaction of flowing fluids with free bodies -- sometimes
compliant, sometimes active, sometimes multiple -- constitutes a class
of beautiful dynamic boundary problems that are central to biology and
engineering. Examples range from how organisms locomote in fluids
(which depends strongly on scale) to how non-Newtonian stresses
develop in complex liquids (strongly dependent on the nature of
fluidic microstructure). I will discuss several interesting examples,
emphasizing how they are formulated mathematically so as to yield
models tractable for analysis or simulation, and show how this work
has interacted with experimental studies.

Partial differential equations on and for evolving surfaces occur in many applications.
For example, traditionally they arise naturally in fluid dynamics and materials
science and more recently in the mathematics of images.
In this talk we describe computational approaches to the formulation and
approximation of transport and diffusion of a material quantity on an
evolving surface.
We also have in mind a surface which not only evolves in the normal direction
so as to define the surface evolution but also has a tangential velocity
associated with the motion of material points in the surface which advects material
quantities such as heat or mass.This is joint work with G. Dziuk