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.

Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, p is an odd prime, and L is a p-extension of K that is Galois over K. Let c be an element of order 2 in Gal(L/k), and H the subgroup of all elements of G := Gal(L/K) that commute with c. Under very mild hypotheses the Parity Conjecture (combined with a little representation theory) predicts that if the rank of E(K) is odd, then the rank of E(L) is at least [G:H]. For example, if L/k is dihedral and the rank of E(K) is odd, then the rank of E(L) should be at least [L:K].

In this talk I will discuss recent joint work with Barry Mazur, where we prove an analogue of this result with "rank" repaced by "p-Selmer rank".