There are a few goals common to all subfields of mathematics, and they are intimately related. One is classification of mathematical objects. This is identical to the entomologist classifying species of insects, can we describe all possible objects that exist?
The best known example is the complete classification of finite simple groups. This classification spans of 10,000 pages of journal articles, but we now know what every single finite simple group in the world is like. Of course, there are non-simple groups and infinite groups, but nonetheless….
Another example is the classification of topological surfaces. This has been done for a specific subset of topological spaces. When complete classification has not been achieved, how do we tell different spaces apart? This is where characterization enters. These are “if and only if” theorems. If we can find a certain property that holds “if and only if” something else holds, we can use one property to discover something new about the other.
Functions are often used to characterize objects. Two groups are considered the same if there exists a function which preserves the group operation. This is called an isomorphism. For example let G, H be isomorphic groups. Then G is abelian if and only if H is abelian. Hence if we have two groups, one is abelian and the other is not, then we know that they are not isomorphic.
It is a little more difficult to distinguish between two topological spaces. For this we assign each topology a set of groups called the “fundamental groups.” Then we can use properties of these groups to tell topologies apart.
The goals of characterization theorems aren’t always so lofty, as this particularly beautiful example I just read about shows. It involves a really boring property of symmetric matrices: negative and positive definiteness. This property arises when attempting to maximize a function of many variables. If the matrix of second partial derivatives (the Hessian), is negative definite at some x, then that x is a local maximum. So clearly it’d be nice to know when a given matrix is negative definite.
One common way of determining whether a matrix is negative definite involves looking at a ridiculous number of determinants related to that matrix—it is very ugly. Another way is an elegant characterization theorem.
This theorem characterizes definiteness in terms of the eigenvalues of the matrix. If all of the eigenvalues are negative, the matrix is negative definite. That’s it. A similar statement holds true for positive definiteness (eigenvalues are positive).
That is beautiful mathematics.
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