Morse Theory of Gradient Flows, Concavity and Complexity on Manifolds with Boundary

This monograph is an account of the author's investigations of gradient vector flows on compact manifolds with boundary. Many mathematical structures and constructions in the book fit comfortably in the framework of Morse Theory and, more generally, of the Singularity Theory of smooth maps.

The geometric and combinatorial structures, arising from the interactions of vector flows with the boundary of the manifold, are surprisingly rich. This geometric setting leads organically to many encounters with Singularity Theory, Combinatorics, Differential Topology, Differential Geometry, Dynamical Systems, and especially with the boundary value problems for ordinary differential equations. This diversity of connections animates the book and is the main motivation behind it.

The book is divided into two parts. The first part describes the flows in three dimensions. It is more pictorial in nature. The second part deals with the multi-dimensional flows, and thus is more analytical. Each of the nine chapters starts with a description of its purpose and main results. This organization provides the reader with independent entrances into different chapters.
Contents:PrefaceAcknowledgmentsFlows in 2D and 3D:A Prelude in 2D: Flows in FlatlandVector Fields, Morse Stratifications, and Gradient Spines of 3-foldsConcavity and Complexity of Traversing Flows on 3-foldsDeformations of Traversing Flows in 3D and Modifications of Gradient SpinesHigh-dimensional Flows:Stratified Convexity and Concavity of Flows on Manifolds with BoundaryTraversally Generic and Versal Flows: Semi-algebraic Models of TangencyCombinatorics of Tangency: the Stratified Spaces of Real PolynomialsComplexity of Shadows and Traversing Flows in Terms of the Simplicial VolumeThe Burnside Ring-valued Morse Formula for Equivariant Vector Fields
Readership: Graduate students and researchers specialized in geometric topology, singularity theory and dynamical systems.Morse Theory;Gradient Flows;Boundary Effects;Convexity;Complexity0Key Features:The text is unique in its subject and approachThe results are novel and have a nice geometrical flavorThe book is richly illustrated to help readers visualize ideas and statements