Singularity Theory

see also:

• https://www.youtube.com/watch?v=0pIRibivolA

Singularity theory is a branch of mathematics that studies spaces of mappings and the singular points of these mappings, focusing on the properties and structures of singularities—places where mappings fail to be differentiable or where differential properties change abruptly. Originating from the study of critical points of differentiable functions and extending to more general settings, singularity theory intersects with algebraic geometry, differential topology, and Complex Analysis, offering profound insights into the behavior of functions and spaces under deformation.

Fundamental Concepts

• Singular Point: In the simplest context, a singular point of a function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a point where the derivative (Jacobian matrix) of $f$ fails to be surjective, leading to a loss of regularity. For algebraic varieties, singular points are where the tangent space does not have the expected dimension.

• Morse Singularities: In the context of Morse theory, a critical point of a smooth function $f:M\to \mathbb{R}$ (where $M$ is a manifold) is considered non-degenerate (or Morse) if the Hessian matrix at that point is non-singular. Morse theory analyzes how the topology of level sets of $f$ changes as one passes through critical values, providing insights into the topology of the manifold $M$.

• Catastrophe Theory: A branch of singularity theory that studies how the qualitative structure of the solutions to equations depends on the parameters. Catastrophe theory classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, modeling phenomena such as bifurcations in dynamical systems.

• Classification of Singularities: Singularity theory seeks to classify singularities up to certain equivalence relations, such as right-left equivalence for maps between manifolds. The classification often involves identifying invariants and normal forms for singularities.

Applications and Examples

• Optics and Physics: Catastrophe theory and singularity theory have been applied to optics to explain phenomena such as caustics and wavefront dislocations, as well as in other areas of physics, where the structure of singularities explains phase transitions and critical points in systems.

• Algebraic Geometry: Singularity theory is integral to the study of algebraic varieties, where singular points on curves, surfaces, and higher-dimensional spaces are analyzed. The resolution of singularities, particularly in characteristic zero, is a significant achievement in this field.

• Differential Equations: In the study of differential equations, especially partial differential equations (PDEs), singularity theory helps understand the formation and propagation of shocks, singular solutions, and other phenomena where solutions fail to be regular.

• Robotics and Mechanism Design: Singular configurations in robotics, where the control or movement of a robot arm becomes problematic, are analyzed using concepts from singularity theory, guiding the design of mechanisms to avoid or leverage these configurations.

Mathematical Formalism

Singularity theory uses a variety of mathematical tools and concepts, including:

• Jet Spaces: Generalizing the concept of Taylor expansions, jet spaces allow for the local study of functions and mappings, providing a framework for analyzing singularities and their unfoldings.

• Whitney Stratifications: A technique for decomposing spaces into manifolds (strata) of varying dimensions, where the singular structure of a space is understood in terms of the geometry and arrangement of these strata.

• Versal Deformations: Understanding how a singularity can be “unfolded” into a family of less singular or nonsingular objects, exploring the space of possible deformations that preserve certain properties of the singularity.

Singularity theory provides a rich and intricate framework for dissecting the nuances of spaces and mappings, revealing the underlying stability, structure, and transitions of systems across mathematics and physics. Its development continues to uncover deeper layers of understanding concerning the behavior of complex systems under variation, influencing a broad spectrum of scientific and mathematical disciplines.