Mónica Clapp: The role of geometry in concentration phenomena

Many models for pattern formation in various branches of science (e.g. biology and chemistry) are based on the idea, proposed by Alan Turing in 1952, that in a system of equations modeling two interactive substances, different diffusion rates could lead to nonhomogeneous distributions of these substances. Often, one of them is highly concentrated in small areas, forming distinctive patterns like narrow peaks or spikes.

Showing existence and determining the profile of solutions for this type of models has been a very active area of research during the last three decades. Many results concerning concentration at a point or at a finite number of points are now available. Quite recently, solutions concentrating at higher dimensional manifolds have been shown to exist.

In this talk I will show how geometry plays a role in producing solutions which concentrate at manifolds of different positive dimensions. This is joint work with M. Ghimenti and A.M. Micheletti.