singularités

Singularités are points in a mathematical or physical system where the usual rules of analysis do not apply. They are often characterized by a singularity in the function describing the system, which is a point where the function becomes infinite or undefined.

In geometry, singularities refer to points in a manifold where the manifold's curvature or other geometric properties become infinite or degenerate. These points can be thought of as "tears" or "folds" in the surface of the manifold, where the usual rules of geometry break down.

In physics, singularities often occur in theories of gravity and quantum mechanics. For example, the curvature of spacetime around a black hole becomes infinite at the singularity at its center. Similarly, in some theories of quantum gravity, singularities can arise from the attempt to calculate the properties of particles at the Planck scale.

Singularities can be categorized into several types, including:

* Coordinate singularities, which occur when the coordinate system used to describe a manifold becomes invalid at a particular point.

* Curvature singularities, which occur when the curvature of spacetime becomes infinite at a point.

* Conical singularities, which occur when spacetime becomes "tesselated" or "patched" at a point, leading to a breakdown in the usual rules of geometry.

Singularities often pose significant challenges in mathematical and theoretical understanding of systems, and have been the subject of much research and debate in various fields, including mathematics, physics, and astronomy.