France/Russia

1999 Balzan Prize for Mathematics

Acceptance Speech – Bern, 16.11.1999

Mrs. President,

Members of the Balzan Foundation,

Ladies and Gentlemen,

The award granted by the Balzan Foundation is a great sign of recognition of Mathematics as an essential ingredient of our culture and the connecting link between different branches of science. I am greatly honored for being chosen to represent the community of Mathematicians as a recipient of the Prize.

Mathematics is the most ancient intellectual enterprise of human beings which cristallized into a logical system around 300 B.C. in Alexandria with the appearance of *Euclid’s Elements*.

Geometry cherished and developed by mathematicians in the course of centuries, turned into a multibranched tree of stunning structural harmony growing out of unforeseen symmetries inherent in the mathematical and physical worlds. The potential of the geometric language, envisioned by Galileo and fully manifested in Newton’s *Mathematical Principles*, remains a powerful drive of modern Physics, from relativity to the String Theory.

The external world framed into geometric space consists of discernably distinct objects, recording and counting of which led us to arithmetic, ultimately revealing an incomprehensively deep and subtle multiplicative structure of numbers, germinating from Euclid’s algorithms and infinity of the primes. And the rules of additivity brought forth the linear analysis and probability theory – the foundations of rational thinking in quantum mechanics, natural sciences, economics as well as in the day-to-day life of the modern society.

Taken broadly, the task of mathematics and mathematicians is to articulate the visible regularities in the physical and mental worlds, and to find new structural patterns unperceivable by direct intuition and common sense.

Let me give a couple of examples close to my domain. Start with the common-sense notion of “distance” which refers to two physical objects, two states of a physical or biological system, or two abstract ideas being mutually close or far apart. A mathematician will speak of an abstract space of the objects in question, called *points* on this occasion, and will think of the distance as a particular assignment of a positive number to each pair of points in the space. We are all familiar with the ordinary 3-space, where the distance between two points is given by the length of the shortest segment between them. However, this would be only of an academic value if you live in a mountain country where a more appropriate distance is given by the length of the shortest available path between two given locations.

A more abstract example comes from mathematical linguistics: our points are sentences, certain sequences of letters, say in the English alphabet, and you want to define a distance between two sentences. The definition, certainly, should depend on the purpose you have in mind. The most naive, yet useful is the *Hamming metric*, where you stick to sentences of a fixed length, say exactly of hundred letters, and measure the distance between two such sentences by the number of locations where the two have different letters.

Now, suppose you are given such a metric space, i.e., a collection of numbers representing the distances. How can one extract the geometric image out of these numbers? For example, you may isolate 10^{3} landmarks on a mountain terrain, and write down an array of 10^{6 }numbers

(actually (10^{6} – 10^{3})/2 as the distance is symmetric and vanishes on identical points) representing the distances. Can you then determine, for instance, the size of the highest mountain by just staring at these 10^{6} numbers? And given two such arrays, can you tell if the two landscapes are similar or quite different?

The latter question brings us to the next level of abstraction: we need to define a distance between given metric spaces. So the spaces themselves are thought of as points constituting a huge abstract object: the space of metric spaces. But abstract or not, we can not help dealing with such spaces, both in pure mathematics and applications. For example, if we want to design a computer program distinguishing human faces or facial expressions, we encode each face by an array of numbers, e.g., thinking of a face as a kind of a landscape with the shortest path distance, and then define, in an intelligent way, some distance between faces. The program must tell us whether two faces, turned into metric spaces, are similar or not. Of course, we can not directly operate with the numbers, e.g., we can not even write down all these numbers in the language space: there are about 26^{200} of them, by far more than the totality of the elementary particles in the Universe! So we must develop a language capable of expressing essential features, called *metric invariants*, of the spaces and then compare spaces according to these invariants.

There is no universal recipe for making up such a language; we try whatever we can, freely borrowing from the geometric intuition based on our experience in the 3-space. This gives us a good start, but we have a long road ahead of us.

The second example is that of the symplectic rather than metric, geometry. This presents itself in the phase spaces of mechanical systems without friction such as the system of three celestial bodies say the sun, the earth and the moon. These are governed by Newton’s laws, where the future is determined by the positions and velocities of the three at a given moment. Thus, the space in question has dimension 18 where each point corresponds to a configuration of possible positions and velocities of the three bodies. Newton’s laws can be encoded into a particular transformation of this space: each point goes to the new one, corresponding to what happens to the bodies after one second of motion ruled by Newton’s laws. This transformation distorts any conceivable distance in the space, it is not at all a “rigid motion” of the 18-space. Yet, it preserves something else, something called the *symplectic structure*. This, unlike the distance associated to the idea of length of a path, is related to the area of surfaces in our 18-space, though it is not the usual area and it is virtually impossible to give an intuitive picture of the symplectic structure as this does not truly expose itself below dimension four. Yet, we want to define symplectic invariants as those are preserved by the above transformation and may eventually tell us something useful about the mechanical systems. The help comes from an unexpected source, the complex analysis: the Cauchy-Riemann equation can be implanted into the symplectic soil eventually yielding a harvest of symplectic invariants. This is purely a mathematician’s game, no direct intuitive explanation, no naive common sense behind it. But the conclusion turns out useful in several branches of geometry, especially algebraic geometry and mathematical physics related to strings and the conformal field theories.

What I have shown you are just two among many green branches on the immense tree of mathematics. All of us who care for the healthy growth of this tree, mathematicians and scientists alike, search for new directions, new challenges trying to encompass all kinds of ideas and reveal the logical structure associated to these ideas.

I believe that a meaningful life of a civilized society much depends on the health of this tree, and I am happy to accept the Prize from the Balzan Foundation, sharing this view and the role and significance of our science.