Mostow rigidity

Imagine you have a really cool Lego model of a building that you built with a lot of effort. You can move the Lego pieces around and change their shape, but you always keep it looking like a building. This is like a special type of shape called a "rigid shape."

Now, let's imagine you have another Lego model of a building with a special feature. This time, the building can bend and change its shape. You can twist the pieces and make it look like a different building. This is like a flexible shape.

Well, in math, there are special shapes called "manifolds" that can also be rigid or flexible. A manifold is like a fancy word for a shape that looks like a normal flat piece of paper, but it can be bent in different ways. Sometimes, these manifolds can be very complicated and it is hard to tell if they are rigid or flexible.

In comes Mostow rigidity, which is a rule that helps us understand if a special type of manifold is rigid or flexible. This rule tells us that if we have a certain type of manifold called a "locally symmetric space," it will always be rigid.

But what's a locally symmetric space? Well, imagine you have a big balloon and you want to draw a pattern on it. But, instead of drawing the same pattern all over the balloon, you want to draw patterns that look the same depending on which side of the balloon you look at. This is what a locally symmetric space is like.

Now, the really cool thing is that Mostow rigidity tells us that if we have a locally symmetric space, we can't change its shape! It is stuck in the shape it was made. Just like the Lego building that can't change its shape once it is built, these locally symmetric spaces can't be changed into different shapes.

So, Mostow rigidity is like a magic rule that helps mathematicians understand and predict if certain shapes called "manifolds" can bend and change their shape or if they will always stay the same. It's a way to uncover the secrets of these special shapes and understand their properties.