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P. Szmolyan:
"Advances in Geometric Singular Perturbation Theory";
Keynote Lecture: Multiscale Phenomena in Geometry and Dynamics, München (invited); 2019-07-22 - 2019-07-26.

Modelling of problems from natural sciences, engineering and life sciences by ordinary differential equations often leads to singular perturbation problems with solutions varying on several widely separated time-scales. The analysis of such systems by methods from dynamical systems theory - most notably from invariant manifold theory - has become known as geometric singular perturbation theory (GSPT). By the efforts of many people GSPT has been applied successfully in the analysis of an impressive collection of diverse problems. Fenichel theory for normally hyperbolic critical manifolds combined with the blow-up method at non-hyperbolic points is often able to provide remarkably detailed insight into complicated dynamical phenomena. Much of this work has been carried out in the framework of slow-fast systems in standard form, i.e. for systems with an apriori splitting into slow and fast variables.

More recently GSPT turned out to be useful for systems for which the slow-fast structures and the resulting applicability of GSPT are somewhat hidden. Problems of this type include singularly perturbed systems in non-standard form, problems depending singularly on more than one parameter, and smooth systems limiting on non-smooth systems as a parameter tends to zero.

In this course I will survey these developments and explain key features in the context of selected applications.