MorseTheory

Introduction

 
 

Morse theory was developed in the 1920s by mathematician Marston Morse. A typical problem in mathematics involves attempting to understand the topology, or large-scale structure, of an object with limited information. This kind of problem also occurs in mathematical physics, dynamic systems and mechanical engineering. Original Morse theory, which applied to a class of mathematical objects called smooth manifolds-such as a plane, a circle and the surface of a sphere-provides a general tools of attacking this problem.

Unfortunately, not all interesting objects are smooth manifolds, however, and there has been a demand for a version of Morse theory that applies to other situations. Some researchers developed such an approach.
In the 1930s, Marston Morse proved a major result that generalize the straightforward result that the lowest-order nonvanishing term in the Taylor series describes the local behavior of a smooth function of single variable (function with derivatives of up to order n) to functions of many variables. As is well-known for any point  x near a given point x 0  the value of function  f  with infinitely many derivatives at x 0  can be expressed by the Taylor series
 ( 1 )                                    f (x) = c 0 (x 0 ) + c 1 (x 0 ) (x - x 0 ) +  c 2 (x 0 ) (x - x 0 ) 2  + . . .
where
                                         c k (x 0 ) = f (k)  (x 0 ) / k!.
To simplify the notation we assume that x 0  = 0 in (1) . In this case (1) becomes:
 ( 2 )                                    f (x) = c 0 (0 ) +  c 1 (0 ) x +  c 2 (0 ) x 2  + . . .

If  c 0 (0) = 0  and c 1 (0 )  different from zero, then (2) suggests that for x sufficiently close to 0, we have
                                          f (x) ~ c 1 (0) x ,
since the higher-order terms (HOT) can all be made as small as we like by taking x near enough to 0. In this situation, we call the origin a regular point of f .
In the case where c 0 (0) = c 1 (0 ) = 0  and c 2 (0 ) different from zero,(2) suggests that for x sufficiently close to 0, we have
                                          f (x) ~  c 2 (0)  x 2  ,
since the higher-order terms (HOT) can all be made as small as we like by taking x near enough to 0. In such case, we call the origin a nondegenerate critical  point of  f .
In the case where c 0 (0) = c 1 (0 ) = c 2 (0 ) = 0  and c 3 (0 ) different from zero,(2) suggests that for x sufficiently close to 0, we have
                                          f (x) ~  c 3 (0)  x 3  .
In such case, we call the origin a degenerate critical  point of  f . One part of Morse's Theorem claims that this case is unusual; nevertheless, it does happen. And when it does, it means that strange things are going on with the physical or biological or sociological  phenomenon that the function represents, such as the buckling of a beam or the outbreak of ecological catastrophe or the outbreak of a political revolution.
Continuing the above reasoning we conclude that the local behavior of  f near the origin is approximated by the first nonzero term in its Taylor series representation (2).
For smooth functions, the key property is how they behave near a critical point since near a regular point the behavior is regular - essentially just that of a straight line or a plane in higher dimensions. So if we want insight into how the function is put together from its various local pieces, we need to find information about its behavior near the critical points. Any critical point of f  - degenerate or nondegenerate - is called a singularity of the function. It is the goal of singularity theory to understand and explain the singularities of a function of one or many variables.
 

Morse's Theorem

 Consider now a smooth function having only nondegenerate critical points. Such a function is called a
Morse function.

MORSE'S THEOREM   Morse functions are stable and dense in the set of all smooth functions. Furthermore, near a critical point of index k there is a smooth change of coordinates under which the resulting Taylor series of the Morse function f  near the origin is the pure quadratic function (form)

f (x 1 ,x 2 , ..., x n ) = - x 1 2  - x 2 2  - . . . - x k 2  +  x k+1 2  + x k+2 2  + . . . + x n 2  .


If f is a function of more than one variable its local geometry near a nondegenerate critical point looks like a saddle, since the graph of the function is a surface that may bend in different directions at a given point. Since the critical point is nondegenerate, this saddle must then curve downward in k coordinate directions and upward in the remaining directions. The integer k, the number of downward curving directions, is called the index of the critical point. 
The first part of Morse's Theorem says that a small, smooth perturbation of a Morse function yields another Morse function. The density means that there is a Morse function arbitrarily close to any non-Morse function. The last part of the theorem says that there is a smooth change of coordinates near the critical point that transforms the function f into a saddle, and that the nature of the saddle is determined by the index k of the critical point.
 

An important application of Morse theory is a catastrophe theory developed by several mathematicians such as R. Thom and others.

Return To Index