% Date: Tue, 29 Sep 1998 12:36:54 +0400 (MSD) % From: Vladimir Arnold % Subject: abstract conference math.2000 (fwd) POLYMATHEMATICS: IS MATHEMATICS A SET OF ARTS OR A SINGLE SCIENCE? by V. Arnold (Abstract of the talk to be given at the conference "Mathematics towards the third millennium" (Rome, May 27-29, 1999) at the Accademia dei Lincei.) The similarities between the ordinary and symplectic differential geometry and topology are sometimes obvious (Poisson brackets of the vector fields and of functions) and sometimes less trivial (general submanifolds in ordinary geometry correspond to Lagrangian submanifold in the symplectic case). The systematic study of the parallelism between different theories lead to many conjectures, some of which turned to be true (and some are now even proved, like those of the Lagrangian intersection theory of Conley-Zehnder-Floer-Chaperon-Laudenbach-Sikorav-Chekanov-Gromov-Hofer and others, providing proofs and the extensions of the "Arnold conjectures" of 1965) The algebraic version of this parallelism is formalized in the theory of root systems. Statements of ordinary linear algebra, once translated into the language of the root system of type A, can be applied to other root systems, providing at least conjectures. These conjectures (which should be sometimes cleverly modified) can be in most cases proved for the general root systems, providing results in Euclidean and symplectic vector spaces (extendable, in most cases also to the "non classical" root systems of type E,F,G and sometimes even to the non cristallographic Coxeter groups). These geometric theories are of course geometries of the vector spaces equiped with some additional structures (Euclidean, symplectic). But the experience of the root system theory shows that it is wiser to consider these theories not as daughters of the ordinary linear algebra (corresponding to the special root system A) but rather as its sisters. The infinite-dimensional counterpart of the classification of the simple Lie algebras is the Cartan's classification of the simple pseudogroups. It contains: 1) the groups of the diffeomorphisms of smooth manifolds, 2) the groups of volume preserving diffeomorphisms 3) the groups of symplectomorphisms 4) the groups of contactomorphisms. There exist also some conformal and complex versions of these teories. The dream of the polymathematics is to transfer statements from each of these theories to the others, guessing this way new results which might be later checked or modified to become theorems. For instance, my work in 1971 on the Hilbert's problem 16th on the topology of real plane algebraic curves was based on the following observation: the complex version of the real manifold with boundary is the two-fold covering (ramified along the complexified boundary). The interrelation between different branches of mathematics that one guesses by this unformal nonaxiomatic way are sometimes strange and unexpected. As an example I might quote the long list of misterious paralell mathematical trinities like: {tetraedron, octaedron, icosaedron} {(60,60,60), (45,45,90), (30,60,90)} {real numbers, complex numbers, quaternions} {Stiefel-Whitney classes, Chern classes, Pontryagin classes}. As an example of a result discovered this way, I mention the following: THEOREM The quotient space of the 4-dimensional quaternionic projective space by the authomorphisms of the quaternionic algebra becomes a 13-dimensional sphere while quotiened by the quaternionic conjugation involution. The complex version of this quaternionic geometry fact is the classical result, known to Pontryagin in the thirties: THEOREM The quotient space of the complex projective plane by the conjugation involution is the 4-sphere. The real version is the obvious: THEOREM The real projective line is the circle. The proofs of these three theorems are exactly the same. They are based on the geometry of the hyperbolic partial derivative equations. The quaternionic version has been discovered while trying to study the quaternionic version of the quantum Hall effect and the Berry phase theory, which might be considered as the complex version of my 1972 "Modes and quasi modes" paper on the topological meaning of the Von Neuman-Wigner theorem on the electron terms repulsion.