Dr. Lisa Randall, a professor of theoretical physics at Harvard, introduces the lecture.

McMullen graduated from Williams College, and went on to study at Cambridge University and the Institut des Hautes Etudes Scientifique in France. He earned his doctorate in Mathematics from Harvard University in 1985. He has won numerous awards, including a Fields Medal, and is a Guggenheim Fellow and a Fellow of the American Academy of Arts and Sciences.

In 2002-2003, out of the blue, Russian Grigory Perelman posted three short papers to the online mathematical bulletin board known as the Arxiv. These papers addressed one of the central problem in three-dimensional space that has puzzled mathematicians for over 100 years. "It's a problem in low dimensional topology – the world of shapes and forms, geography and maps, coffee cups and donuts, loops and knots," said McMullen. Today, 3 years later, a consensus has developed that the work is correct and that the problem is solved.

To explain this breakthrough, McMullen begins with "the paradoxical problem of the shape of the Earth." The earth is finite, but it has no edge. Travelers leaving from the North Pole meet again at the South Pole, and a traveler walking straight north from Cambridge could walk all the way up to the pole, back down the other side to the south pole, then back north to Harvard Square without stopping. But a ball is not the only shape that is finite and has no edge: another, radically different shape is a donut, or torus.

One way to sense the global difference between a ball and a donut is to apply the Loop Test. No matter where you tie a loop of string around a ball, you can pull the loop tighter and tighter around smaller and smaller parts of the sphere until the string encloses a single point (or falls off!). When you tie a loop around a donut, you cannot pull the string any tighter without cutting through the donut; it fails the loop test.

Mathematicians classify surfaces by the number of holes or "handles" they have; this number is called the genus. A sphere has genus 0; a donut has genus 1. Topologically, an animated shark is the same as a sphere; they both have genus 0. As the genus increases, shapes fail the loop test more and more dramatically. Spheres are the only closed surfaces that pass the loop test.

All these surfaces, no matter what the genus, can be built out of simple tiles; the question is, which type of geometry gives you the best tiles for covering the surface? Euclidean geometry works for shapes in the usual flat plane. Spherical geometry deals with how circles look on spheres, like the "great circle" routes that jets use to travel the globe. Hyperbolic geometry is based on the hyperbolic plane, a disk which grows exponentially larger in all directions. "There's an enormous amount of space toward the edge of the hyperbolic plane – and it's this explosion of room that makes it such a versatile geometry or architecture for building more complicated surfaces," McMullen said.

Genus zero surfaces, like the shark and the sphere, can be built using spherical geometry. Surfaces of genus one – tori – can be built using Euclidean geometry. "The torus can be neatly tiled by a network of squares without there being any overlap, and with the tiles all fitting together perfectly," McMullen said. Alternatively, a single square can be wrapped up into a cylinder; join the cylinder's ends, and you have a torus.

A genus two surface is trickier. A shape with two holes can be built by gluing together eight five-sided tiles. To fit together cleanly, those five-sided tiles need straight edges and 90-degree corners that will let them fit together at a vertex. To find a pentagon with 90-degree angles, mathematicians look to hyperbolic geometry. "If you had enough of these pentagons, you could build any surface of any genus two or more than two," McMullen said.

How can the universe be finite but have no edge? Perhaps the universe is a 3-sphere, a closed space where "everything that rises must converge" to a single point – a system which Dante envisioned culminating in the "Empyrean" in his 14th century Divine Comedy. "It might seem paradoxical that rockets flying away from earth in all different directions should ultimately reconverge and all land at this 'Empyrean', at the ceiling of heaven," said McMullen, "but it is no more paradoxical than the fact that travelers leaving from the North Pole ultimately reconverge and meet again at the South Pole," McMullen said.

Just as the Earth's surface might be either a ball or a donut, the Universe might be shaped like a 3-sphere or a 3-torus. And just as for surfaces, the 3-sphere passes the Loop Test, while the 3-torus does not.

The Poincaré Conjecture, formulated by a French mathematician in the 1890s, states that the only three-dimensional space which passes the Loop Test is the 3-sphere. "Another way of saying this is that there is no fake 3-sphere," said McMullen.

To prove the Poincaré Conjecture, we need to show that something does not exist - that there are no other manifolds that pass the Loop Test besides the 3-sphere. The Four Color Conjecture is similar in spirit.

The Four Color Map Conjecture states that using just four hues, you can paint any map so that adjacent countries have different colors. To approach the proof, Wolfgang Haken and Kenneth Appel considered the possible shapes of the simplest map that really needs 5 colors. Using computers to analyze thousands of cases, they showed that one could always eliminate one or more countries and get a simpler map that still requires 5 colors. This is a logical contradiction, so four colors suffice.

One could try, in the same way, to study the simplest fake 3-sphere and show it could be made yet simpler. Luckily this proof, which would require enormous reliance on computers for verification, has never been carried out.

Thurston proposed to attack the Poincaré conjecture by studying an even stronger assertion, the Geometrization Conjecture, which states that every 3-manifold can be built using one of just eight styles of architecture. This is a better question because it applies to all 3-manifolds. Spherical geometry is the only one of those eight architectures which can build a space that passes the loop test – and so the Geometrization Conjecture implies the Poincaré Conjecture.

Richard Hamilton proposed to attack the Poincaré and Geometrization Conjectures not by tiling space, but by transforming it. Hamilton's method is called Evolution by Curvature. "The idea is to take our unknown or distorted manifold and gradually remold it back into a recognizable shape," McMullen said. Hamilton uses the Ricci Tensor from general relativity, a measure of the manifold's curvature, to guide the evolution and make the manifold more symmetric.

It's a powerful but mysterious technique. Likening topology to word problems, McMullen said, "Hamilton proposes to solve a crossword puzzle using an atom smasher." Evolution by curvature aims to solve a global, topological problem by purely local considerations. "Each of these points is moving according to a law that has nothing to do with what any of the other points are doing. No one is examining the global geometry... it's as though a million tiny ants working independently could undo a knot in your garden hose," McMullen said.

What makes this approach challenging is that as the manifold evolves, it can develop singularities; it can split into "droplets" which evolve independently. The singularities are not a bad thing – they can help split apart pieces of a manifold which have incompatible architecture and cannot be analyzed while they are joined – but an analysis where singularities can occur is very difficult.

Perelman's earlier works include research on Alexandrov spaces – just the sort of spaces that can arise as limits to evolution by curvature. Perelman took Hamilton's work three steps further. He showed that singularities are always "undoing tunnels" between incompatible architecture. Then, he came up with a process for smoothing out the “corners” of the singularities, called “evolution with surgery.” With two smooth pieces, Perelman proved, the evolution with surgery can be continued for all time and eventually the architecture of the underlying manifold becomes visible.” Perelman's three papers prove that the evolution by curvature process reconstructs the architecture of the underlying manifold. Consequently, the Geometrization Conjecture is true, and consequently the Poincaré Conjecture is true,” said McMullen.

This proof doesn't mean that we have a simple catalog of all possible 3-dimensional spaces. “It's not as easy to write a list of spaces as we did before by just counting the number of holes,” said McMullen. However, it does mean that all 3-dimensional spaces can realized as rigid geometric objects; that these objects can be constructed from one of eight reputable architectures. "There are no mysterious spaces like a fake 3-spheres that are lurking on some sort of black market," said McMullen.

Perelman based his proof on other mathematicians' work, and many mathematicians have worked on expanding Perelman's work into detailed and complete proofs over the last few years. "Dieudonné called mathematics 'the music of reason'," said McMullen. "The analogy is apt if you allow that a great opera can take 100 years to write. What's remarkable is that the end result is not a cacophony; rather, it is a masterpiece beyond the reach of any single composer."

Q. I was told that entropy plays a role in this proof. Is that correct?

Q. A few years ago there was a book called The Millennium Problem by Keith Devlin and he talked about the Poincaré Conjecture. When he talked about classification of 2-manifolds, he talked about two ways of making a new way – what is your way of changing the genus? The other was by putting a cap on a sphere and Möbius strip, some fanciful way of attaching it. Is that different from what you are talking about?

Q. I’m wondering about the integer quality of genus, for example if you had a tetrahedron and removed one triangle. Could that be a shape of genus one-half?

Q. So, I’m perfectly happy for this to be an exploration of abstract reasoning. I think it’s very interesting, but as a physicist and physical chemist, I’m curious because Ricci Flow has an analogy to physics. Are there obviously useful things in physics that this might suggest?

Q. You mentioned the question whether the world we live in, the cosmos, is an actual sphere or possibly not simply connected. What would be the distinction if the decision falls one way or the other, and some physical implications?

Q. I wondered, what happens at still higher dimensions?

Q. How dense was Perelman’s work? I know it extended from his slightly under 100 pages into 300+ pages in the Journal of Mathematics. A lot of work it took 3+ years, so how dense was it originally?

Q. The question a couple of questions ago didn’t really need much hope for this book. Can you do the band test and either give and define your object in a case of an infinite dimensional space? ... So it actually gives you some “freedom.” ... Of course I’m imagining at all how we can do the band around the torus in infinite dimensions, so I can kind of understand this sphere.

Q. Was there something unusual about Perelman posting on the Web? Was he not sure he would get published somewhere else? Is there a chance if he had done this 15 years ago, he will have still been stuck?