tags: - colorclass/differential geometry ---see also: - Invariants - Bifurcations - Bifurcation Theory
Whitney stratifications are a fundamental concept in differential topology and singularity theory, introduced by Hassler Whitney in the 1960s. They provide a systematic way to decompose a space, often a singular space, into manifolds of varying dimensions, called strata, that fit together in a regular way. This decomposition allows mathematicians to extend the tools and intuitions of smooth manifold theory to more general spaces, including those with singularities.
Definition and Key Properties
A Whitney stratification of a topological space is a partition of into a locally finite collection of disjoint, connected, smooth manifolds (called strata) that satisfy certain regularity conditions (Whitney’s conditions A and B) to ensure a controlled behavior of strata near each other.
- Strata: Each piece in the decomposition, or stratum, is a smooth manifold that can vary in dimension. The strata are subject to certain conditions that prevent “bad” interactions between different strata.
- Whitney’s Condition A: If a sequence of points in a stratum converges to a point in a lower-dimensional stratum , and if the tangent planes to at converge, then the limiting tangent plane contains the tangent plane to at the limit point.
- Whitney’s Condition B: Given two sequences of points in and in converging to the same point in , if the secant lines (lines connecting to ) converge to a limiting line and the tangent planes to at also converge, then the limiting line lies in the limiting tangent plane.
Importance of Whitney Stratifications
Whitney stratifications allow mathematicians and scientists to:
- Understand Singular Spaces: They provide a framework for analyzing spaces with singularities, such as Algebraic Varieties, by decomposing them into simpler, well-understood pieces.
- Extend Differential Topology Tools: Tools from differential topology and geometry can be applied to each stratum of a Whitney stratified space, enabling the study of spaces that are not smooth manifolds in their entirety.
- Analyze Dynamical Systems: In dynamical systems, Whitney stratifications can be used to study the behavior near singularities and invariant sets, such as attractors or repellers.
Applications
Whitney stratifications have found applications across several areas of mathematics and related fields:
- Algebraic Geometry: In the study of algebraic varieties, stratifications help in understanding the local and global structure of varieties, including their singular and nonsingular parts.
- Singularity Theory: Stratifications are crucial in the analysis of singular points of mappings, offering a way to classify and study different types of singularities.
- Dynamical Systems and Physics: They are used to analyze the phase space of dynamical systems, especially in understanding the structure of chaotic attractors and the bifurcation of invariant sets.
- Optimization and Data Analysis: Stratifications have been applied to optimization problems and data analysis, where the geometry of the problem or the data set may naturally lead to a stratified structure.
Mathematical Challenges and Developments
The theory of Whitney stratifications involves deep mathematical concepts from differential geometry, algebraic geometry, and topology. One of the challenges is to construct a Whitney stratification for a given singular space, a task that can be highly nontrivial depending on the complexity of the space. Recent developments continue to explore the connections between Whitney stratifications and other areas, such as Tropical Geometry and symplectic topology, broadening the applicability and understanding of these stratifications in mathematics.