Question

All my life I do topological classification of Morse-Smale systems (flows and cascades). Today, fundamental science is not in fashion, to receive grants require an application. Can you please tell where to look.

Answer 1

I know this is an older question, but since it may interest some, here are some papers about Morse-Smale theory applications:

• This paper describes applications of Morse-Smale theory to sensor networks. It has great pictures, and describes how Morse-Smale theory allows one to avoid slow points in computer computations, among other benefits.
• This paper describes applications of Morse-Smale theory to video segmentation. In part, it says,

In computer vision, detail has been often identi- fied with scale, and so-called scale-space approaches have achieved simplification by blurring. However, blurring obliterates small detail together with the boundaries of large regions. A more discriminating treatment of scale would instead eliminate small detail while keeping large parts crisply delineated. In this paper, we propose a representation of video that captures structure and affords flexible control of detail, without sacrificing crispness. Specifically, we represent structure through the so-called Morse- Smale complex of a scalar function f(x,y,t) associated with the video data, and we obtain a hierarchy of increasingly simple complexes through the process of topological simplification.

• This thesis also describes how Morse-Smale theory can be used to simplify data storage.

All of these papers have references to other papers doing similar things. It seems like Morse-Smale theory has very practical applications to real-world issues!

The main ideas in these papers seem to be that having transverse stable and unstable manifolds gives a nice cell structure or helps us find saddle points more easily.