EDWARD WITTEN

I'd really like to offer a few inspiring words today, but probably the real inspiration here is for me, having the chance to participate in this event with so many of the most talented mathematical high school students in the world. You've all had

such a wonderful mathematical education in getting the opportunity to prepare for and participate in this competition. It makes me think a little bit of my own mathematical education as a youth but without going into any details there, let me just say that

I think it's wonderful, the opportunities that you've had.

Now living as we do at the dawn of the 21st century, we know that the continents have been explored, and in many areas of endeavor which presented challenges hundreds of years ago, the same challenges aren't there today. But the one word of inspiration to today's young people that I feel most strongly about, is that in mathematics and in many areas of science, certainly in the areas of theoretical science that I know best, the challenges are just as wide and just as exciting as they ever were in past generations. Nor are the problems going to be solved quickly so not only for you but as well for your younger siblings, you needn't worry that all of the problems will be solved before you get there.

Now I thought that today, I was going to give you a small taste about the area of science that I work in, which is mostly quantum field theory and the effort to understand the basic structures of nature. The mathematical connection arises because the laws of nature have turned out to rest upon amazingly subtle mathematical structures that have been discovered over the centuries and which to a large extent seem to be still unknown mathematical structures that belong to this coming century.

So I've tried as much as I can in a single page here to summarize what the basic laws of nature were reduced to in the twentieth century. In the 20th century physicists distilled our understanding of nature down to two broad theories: there's quantum mechanics, which is the theory of atoms, molecules and the subatomic world. It's an amazingly rich and subtle theory which, when it was discovered in the 1920's, seemed to

some people like it was an incredibly beautiful yet utterly impractical theory. But ultimately it became needed by engineers and practical-minded physicists in a host of areas: superconductivity, transistors, condensed/solid state physics and many, many other areas. In terms of the physics involved in this theory, one principle - a very strange principle - that many of you will have heard of is the Heisenberg uncertainty principle, which asserts that many of our familiar concepts like the position and velocity of a particle, become fuzzy in the subatomic world. In terms of the math involved, it involved many surprising mathematical structures like Hilbert spaces that physicists had never expected to meet but which turned out to be central in the subatomic world.

While quantum mechanics is our basic framework for understanding the micro-world, the world of individual particles, protons, neutrons, electrons, and so on, our basic theory for understanding the universe at large is Einstein's theory of gravity, his greatest achievement, which is called general relativity. General relativity is Einstein's conception according to which gravitation results from the curvature of space and time, much like if you put a mass on a mattress, the mattress is bent, and a marble on the mattress will follow a curved path around it. Einstein's conception was that the sun distorts the space around it, and the planets in orbiting the sun are simply attempting to follow straight lines in the curved world around the sun. So these were the two most basic theories of 20th century physics, and really the greatest achievement of the 20th century was that a wealth of knowledge in previous theories was extended and distilled down to these two broad structures. Now quantum mechanics and general relativity both describe part of the same natural world so it must somehow be possible for them to work together. The strange thing though is that when you attempt to combine them, you run into hopeless contradictions, or at least in a straightforward attempt to combine them you run into hopeless contradictions. The nature of the contradictions is that the nonlinear mathematics of Einstein's theory, based as it is on the 19th century Riemannian theories of curved spaces, clashes with the requirements of quantum theory. Now this problem was first perceived generations ago, and it's been pursued from many angles for the last 60 or 70 years. But there only is one approacj which has achieved anything that seems significant, and it was not found by someone sitting in an ivory tower and emerging seven years later, with a brilliant inspiration about a new theory of quantum gravity. That worked as we all know for proving Fermat's last theorem, in the work of Andrew Wiles from whom you've just heard, but physicists instead stumbled by accident on a trail that led them to what seems to be a new framework of physics that does reconcile quantum mechanics with gravity.

This is the theory known as string theory where an elementary particle is not a point object as I've sketched on this page on the left, but instead is replaced by a loop, a vibrating string. Now you all know that a string can vibrate in many ways. The whole richness of music for example comes from the fact that a piano string or a violin string, or the note played on a flute is not a pure frequency but contains many higher overtones which are different forms of vibration of the same string. It's because of the richness of those higher overtones that we have symphony orchestras and not just tuning forks, which produce the pure note, but sound hopelessly harsh to the ear. So just as the higher overtones lead to the beauty of music, in string theory, the string can oscillate in many

different ways, giving rise to the many different particles and forces that we see in nature. Although we don't know for sure that this theory is right, in the context of this theory, the unity of the forces is achieved by interpreting the different particles as higher overtones or different forms of vibration of a string. Now 20th century physics was strange, but

if string theory is correct, it introduces many equally strange and fundamental new ideas in the picture.

I've tried to illustrate this by this little magic square where in the upper left I draw a picture that's meant to symbolize a conception based on Einstein's special relativity, dating to the early 20th century, of a massless particle traveling at the speed of light in spacetime. In the picture, space runs horizontally, time runs vertically, and the massless particle travels at a 45 degree angle which is meant to illustrate propagation at the speed of light. Now this is a very naive idea, and Einstein elaborated it into the nonlinear wave equations for which he's really famous �\ the Einstein equations which assert that the Ricci tensor of spacetime is zero. These equations encode Einstein's amazingly profound mathematical insight describing just how the curvature of the universe leads to

gravitation. So Einstein gave one profound generalization of the naive idea of a light ray in spacetime, which I show in the upper right corner of the magic square, but going down to the lower left corner, there is a different generalization of the idea of the light ray, which is the oscillating string. It's a completely different way of taking off from the simple idea of the light ray. One direction leads to the Einstein equations, and the other direction leads to the vibrating string. Somehow what string theorists are trying to do is to combine the notion of the vibrating string with the notion of the Einstein equations and so to give what I've placed in the lower right hand corner, where I've only indicated two question marks, perhaps to be filled in thirty or fifty years hence, by

one of you or by one of your children who will discover the successor to the Einstein equations that are related to the vibrating string the way the Einstein equations are related to the point particle.

Some real progress was made in the 1990's but as so often, it didn't occur in the direction that people suspected. If you had taken a poll among theoretical physicists at the beginning of the 90's about where progress would occur in that decade, practically everybody would have been wrong. People would have given their laundry lists of famous problems that they most wanted to see solved, but what you get is often not what you guess, or what you might have thought you wanted. At the beginning of the decade,

we were in the situation that there were five conceivable string theories, plus a wildcard, 11-dimensional supergravity. This was tremendous progress compared to the standard framework of physics, where there were infinitely many possible quantum field theories. But it still left you with the puzzle, if one of these five theories describes our universe, who lives in the other four worlds? Well that question was actually addressed in the

90's when it was discovered that all the possible theories were limiting cases of one, still mysterious theory.

This was done by getting a better understanding of how to combine quantum fuzziness with the notion of the vibrating string and learning that when we took quantum mechanics into account, the five string theories were different limiting cases of one richer theory. So by now we've understood after 30 years of exploration of string theory

that there's really only one candidate for super-unification. It is one theory that physicists have been studying from many different points of view for 30 years but without really understanding what it is. Only in recent years have we come far enough to understand that there was one elephant of which some physicists were studying the trunk, some were studying the tusks, and some the tail and so on as in the famous parable or children's story but that there really was one elephant that we were studying from different points of view. We�ve gotten this far, but still it�s all so dimly lit that there's a whole world to explore.

In terms of experiment the world to explore is, above all, to explore the notion of supersymmetry, which this theory has given birth to. We hope supersymmetry can be proved at the next generation of particle accelerators that will be operating in the coming decade. Those of you who are interested are definitely on time to work at those accelerators in their glorious days, as graduate students or after receiving your PhD's. And on the purely theoretical side, we know that there is a new mathematical description needed because, unlike Einstein's problem where the pre-existing theory of Riemannian geometry proved to offer the right mathematical framework for what Einstein was trying to do, here it seems that the proper framework not yet known. It seems that there is a new mathematical structure lying behind this theory that physicists have grappled with for the last 30 years, and still don't understand. So there's a whole world to explore, and I�d like to say to the 500 young people present at this Olympiad as well as all the other talented young mathematicians in your host countries, there's lots to do both as theorists and experimentalists. Thank you.