chapter 1 Sturm Theorems for Linear Parabolic Equations and Intersection Comparison.-equations -- chapter 2 Transversality, Concavity and Sign-Invariants. Solutions on Linear Invariant Subspaces -- chapter 3 B-Concavity and Transversality on Nonlinear Subsets for Quasilin- ear Heat Equations -- chapter 4 Eventual B-convexity: a Criterion of Complete Blow-up and Extinc- tion for Quasilinear Heat Equations -- chapter 5 Blow-up Interfaces for Quasilinear Heat Equations -- chapter 6 Complete and Incomplete Blow-up in Several Space Dimensions -- chapter 7 Geometric Theory of Nonlinear Singular Parabolic Equations. Max- imal Solutions -- chapter 8 Geometric Theory of Generalized Free-Boundary Problems. Non- Maximal Solutions -- chapter 9 Regularity of Solutions of Changing Sign -- chapter 10 Discontinuous Limit Semigroups for the Singular Zhang Equation -- chapter 11 Further Examples of Discontinuous and Continuous Limit Semi- groups.