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P. Szmolyan:
"Geometric singular perturbation theory beyond the standard form";
Keynote Lecture: Dynamics Days Europe 2019, Rostock (invited); 2019-09-02 - 2019-09-06.

Due to the efforts of many people geometric singular perturbation theory (GSPT) has de- veloped into a very successful branch of applied dynamical systems. GSPT has proven to be very useful in the analysis of an impressive collection of diverse problems from natural sciences, engineering and life sciences. Fenichel theory for normally hyperbolic critical man- ifolds combined with the blow-up method at non-hyperbolic points is often able to provide remarkably detailed insight into complicated dynamical phenomena, often even in a con- structive way. Much of this work has been carried out in the framework of slow-fast systems in standard form, i.e. for systems with an a priori 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. Often several distinct scalings must be used to cover the dynamics of interest and matching of these different scaling regimes is carried out by the blow-up method.
In this talk I will survey these developments and illustrate their relevance in the context of selected models from cell biology.