MORSE FUNCTION Meaning and
Definition
-
A Morse function is a mathematical concept used in the field of differential topology and differential geometry. It is named after mathematician Marston Morse, who introduced the idea in the early 20th century.
In simple terms, a Morse function is a smooth real-valued function defined on a smooth manifold. It possesses certain properties that make it useful for studying the topology of the manifold. Specifically, a Morse function has non-degenerate critical points, which are points where the derivative of the function vanishes. These critical points classify the topology of the manifold by providing information about its critical values and their index, which measures the behavior of the function near that particular critical point.
The concept of a Morse function is particularly important in Morse theory, a branch of mathematics that explores the relationship between critical points, the behavior of the function, and the topology of the underlying manifold. By studying the critical points of a Morse function, mathematicians can extract valuable information about the structure and properties of the manifold, such as the number of handles or holes it possesses.
Morse functions find applications in various fields, including differential geometry, algebraic topology, and theoretical physics. They provide a powerful tool for understanding and classifying the topology of manifolds, and many results in these fields rely on the use of Morse theory and Morse functions.
Etymology of MORSE FUNCTION
The term "Morse function" is named after the American mathematician Marston Morse (1892-1977), who made significant contributions to the field of differential topology. In 1925, Morse introduced a powerful technique called Morse theory, which studies the topology of smooth manifolds by examining critical points of special functions on these manifolds, now known as Morse functions. These functions are smooth, real-valued functions such that the values of the function decrease and increase as one moves away from the critical points. The etymology of the term "Morse function" therefore originates from Marston Morse's influential work in the development and application of Morse theory.
