Equivariant cohomology

Equivariant cohomology is a way of understanding certain mathematical objects, called spaces, that have a special type of symmetry. This means that if you do something to the space, like rotating it or flipping it, it looks the same as it did before.

To understand equivariant cohomology, imagine you have a toy that can be put together in different ways to make different shapes. Each shape has a certain number of different ways it can be flipped or turned, but if you take a picture of it, the picture will always look the same. This is like the symmetry of a space that equivariant cohomology tries to understand.

The idea behind equivariant cohomology is to look at how the space behaves under these different symmetries. If you take a picture of the toy from one angle, it might look different than if you take a picture from another angle. Similarly, if you look at a space from the point of view of a certain symmetry, it might look different than if you look at it from a different symmetry.

Equivariant cohomology gives us tools to study these differences and how they relate to each other. In other words, it's like looking at the toy from different angles to get a better understanding of how it works. This can help us answer questions like "how many different shapes can you make by putting the toy together in different ways?" or "how do the different symmetries of the space affect its properties?"

Overall, equivariant cohomology is a way of using symmetry to better understand mathematical objects, like spaces, which can be described and studied in many different ways. It helps us get a more complete picture of how these objects work, by looking at them from different perspectives.