Recent topics in pattern formation theory

Two recent topics arising in pattern dynamics are presented. One is a characterization of amorphous structures by using computational topology, especially persistent diagrams that extract the information of size and shape of various holes embedded in amorphous structures. The other is about an organizing center for the pulse generators arising in reaction diffusion systems with heterogeneities. Although the kinetics is of the excitable type, pulses can be produced spontaneously without external force from the existing heterogeneities in the media. It turns out that a double homoclinic loop to an unstable pattern created by the heterogeneity is responsible for the generating mechanism through the intensive search of all relevant solutions. These two examples show that huge reduction of information is possible via two mathematical methodologies.