Reduction methods for quasilinear differential-algebraic equations

Geometric reduction methods for differential-algebraic equations (DAEs) aim at an iterative reduction of the problem to an explicit ODE on a lower-dimensional submanifold of the so-called semistate space. This approach usually relies on certain algebraic (typically constant-rank) conditions holding at every reduction step. When these conditions are met the DAE is called regular. We discuss in this contribution several recent results concerning the use of reduction techniques in the analysis of quasilinear DAEs, not only for regular systems but also for singular ones, in which the above-mentioned conditions fail.