I just finished a wondeful book, The Poincare Conjecture: In Search of the Shape of the Universe by Donal O'Shea, and because I'm not very good at writing I'm just going to quote a good chunk:
Mathematics reminds us how much we depend on one another, both on the insight and imagination of those who have lived before us, and on those who comprise the social and cultural institutions, schools and universities, that give children an education that allows them to fully engage the ideas of their times. It is up to all of us to ensure that the legacy of our times is a society that stewards and develops our common mathematical inheritance. For mathematics is one of the quintessentially human activities that makes us more fully human and, in so doing, leads us to transcend ourselves.
Looking up at the night sky, at the distant stars and galaxies and clusters of galaxies, it is inconceivable to me that there are not other intelligences out there, some far different then us. Hundreds of years hence, if we ever develop technologies that enable us to meet and to communicate, we will discover that they will know, or want to lknow, that the only compact three-dimensional manifold in which every loop can be shrunk to a point is a three-sphere. Count on it.
There were lots of mathematical concepts in this book that I didn't understand, but these two paragraphs I surely understood.
If you don't want to read the book but want to understand what the conjecture is, just watch this video: