This term has multiple meanings, depending on context.
- Mathematics Term -
Refers to a point where a function goes to infinity. For a function defined on the complex plane, an essential singularity exists at a point if and only if, for any real number R and complex number Z you take, there will be a point no farther than R from the point of essential singularity for which the function has the value Z. In other words, no matter how small a region around the point of essential singularity you take, you will still find all of the complex numbers inside it as values of the function.
- Astrophysics Term -
The term singularity is used in astrophysics when a mathematical models used to predict and/or calculate a certain state of matter, space-time or even the universe, breaks; yielding infinitly large numbers. This point is called a singularity, since its mathematical form is no longer valid.
A singularity exists in the big bang model of the universe. In this model, at some point, the density of the universe and the curvature of space-time would have been infinite. This means that even if there were events before the big bang, one could not use them to determine what would happen afterward, because predictability would break down at the big bang. This is called the Big Bang Singularity.
Scientists are in search for a new theory that would unite the two main theories used to describe the universe - the general theory of relativity and quantum mechanics. The new theory - quantum gravity - is expected to predict what actually occurs at a singularity.
A special type of singularity was conjectured and proved to exist is the Naked singularity.
- Futurist Term -
Inspired by Moore's Law, futurists have speculated about the imminent arrival of a technology singularity, in which technological progress accelerates toward infinity. This was first predicted by Vernor Vinge in his famous essay Singularity, published in 1993:
"Within thirty years, we will have the technological means to create superhuman intelligence. Shortly thereafter, the human era will be ended."
Full article can be found at: http://www.ugcs.caltech.edu/~phoenix/vinge/vinge-sing.html
Prominent theorists/speculators on the subject include:
Some key concepts are: