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Mathematics Colloquium – Dr. Ryan Goh

Author: las-digital

Patterns and growth

Externally mediated, or “grown,” spatial patterns have become a topic of recent
interest in many fields, such as directional quenching in alloy melts, growing
interfaces in biological systems, and traveling reaction fronts. Here researchers are
interested in how a spatially progressive growth process can control and select
patterns formed in the wake. Mathematically, such growth processes can be encoded
in a step-like parameter dependence that allows patterns in a half plane, and
suppresses them in the complement, while the boundary of the pattern-forming
region propagates with fixed normal velocity.
In this talk, I will show how techniques from dynamical systems, functional analysis,
and numerical continuation, can be used to study the effect of these traveling
heterogeneities on patterns left in the wake; finding for example how the speed of
growth influences the orientation and deformation of striped patterns. I will explain
this approach in the context of the Swift-Hohenberg equation, a prototypical partial
differential equation model for many pattern forming systems, posed in one and two
spatial dimensions.