What is the shape of space? This is one of the questions that has most intrigued me for the last 30 years, since a summer afternoon that has remained etched in my memory. I was a teenager, reading a popular encyclopedia of astronomy in my sun-drenched garden. At the end of the book, some more technical pages introduced the ideas of general relativity and curved space. I understood nothing, but I was fascinated. A certain Albert Einstein had shown that space and time were not as simple as was suggested by our geometric intuition. One statement above all piqued my curiosity: it said that, in a gravitational field, the space-time continuum was no longer Euclidean, but became a referential mollusk. A very clear image formed right away in my mind. To the flavor of the words was added their metaphorical value: in my imagination the space-time mollusk gave birth to a picturesque vision of an immense cosmic snail, its skin streaked through with light, variegated in bends and curves. From then on, I have never stopped seeking to clarify this strange assertion-what is this universal mollusk?—and I have never again looked the same way upon the beautiful starry skies of my native Provence. It was no longer the myriad of stars, flowing through the Milky Way like rivers of diamonds, that intrigued me, but that which surrounded them, space. It was no longer the contents but the container that caused questions to rush forward: does this impalpable space, which contains the stars, have a structure? Is it flat, dented, curved, folded, smooth, rough, grainy? Is it finite or infinite? Does it

have boundaries, holes, handles? And then, what does it mean, exactly, to say that space has a shape?