Research

One of the most common questions I get from non-mathematician is whether mathematicians keep making new formulas. This is a natural impression. In K-12 mathematics, and even in some college mathematics courses, we mostly learn mathematics through arithmetic and calculation. We are given a problem and try to find the answer. But mathematics is not only about calculation.

 Mathematicians study many kinds of objects, and one of the main things we study is the relations between those objects. There are many kinds of relations. One is an order relation. Another is a function, which describes how inputs in one object are sent to outputs in another. Another is an equivalence relation, which tells us when two objects should be regarded as the same.

For example, two triangles of the same size and shape are congruent, even if they are drawn in different places. If their sizes are different, but one can be obtained from the other by scaling, then they are similar. If their sizes and angles are both different, we may still call both of them triangles.

In each case, we are deciding which differences matter and which differences we want to ignore. This is one aspect of modern mathematics: we look for common properties among different objects, define a notion of when two objects should be considered equivalent, and then try to classify the resulting types.

Order relations appear just as often. For real numbers, any two numbers can be compared. Sets have a different kind of order. If every element of a set A is contained in a set B, we write A⊆B. But unlike real numbers, two sets do not always have to be comparable. It may happen that neither A⊆B nor B⊆A. This makes a familiar construction look a little different. Given two sets A₁ and A₂, we can ask for the smallest set that contains both of them. The answer is their union, A₁∪A₂. So the union of two sets can be understood as an answer to a question about the order relation on sets: among all sets larger than both A₁ and A₂, find the smallest one.

Functions are another basic way of describing relations between mathematical objects. In school mathematics, a function is often presented by a formula such as f(x) = x². But a formula is only one way to describe a function. More generally, a function tells us how elements of one object correspond to elements of another. We can study functions between any sets and many other mathematical objects.

From this point of view, much of mathematics is not about finding numerical answers. It is about understanding objects through the relations between them.

What does Topology studies?

Topology studies geometric shapes, often in high dimensions, and asks when two of them should be regarded as equivalent. One of its main goals is to understand the equivalence classes that arise from this relation and to classify them. We often study properties that do not depend on distance or size. For example, the perimeter and area of a triangle depend on its size. On the other hand, the fact that the sum of its interior angles is always 180 degrees does not.

Topology goes one step further. We allow a triangle to be continuously deformed into a square, a pentagon, or even a circle, and regard all of them as the same shape. This very flexible notion of equivalence turns out to be especially useful when studying geometric objects in higher dimensions.

To do this, topologists use the other kinds of relations as well. We can break a space into smaller subspaces and study how they are ordered by inclusion. We can also study functions between spaces and use them to compare one space with another.