(Steklov Institute, Moscow and Universite Paris-Dauphine)
"Wave propagation theory, Legendrian knots and contact topology"
One of the simplest applications of the theory which we shall discuss
goes back to the use of light ray caustics to burn objects (which
Aristophanes attributes to Socrates and which was later used, it
seems, by Archimedes to destroy the Roman's navy).
Consider a closed curve on the Euclidean plane (say, an ellipse). Suppose that waves are propagating inside the domain bounded by the curve, the propagation velocity being equal to one. One can see that at some moment the wave front curve will have 4 cusp point singularities (of the semicubical type).
One of the recent theorems of the theory of Legendrian knots (that I shall discuss in this talk) claims that these four cusps are necessary in every problem of wave propagation theory (for instance, the propagation velocity might depend on the point of the plane as well as on the time moment).
The relation between wave propagation theory and knot theory is unexpected and was discovered only a few years ago. It follows from Huygens'principle, that is from the wave-particle duality. The rays are described by Hamilton's system of ordinary differential equations. The resulting absence of intersections of rays provides the persistence of the topological type of the knot, defined by the evolving wave.
The fact that a function on a circle has at least two critical points, is a particular case of the Sturm theorem. Generalisations include the proofs of the so-called Arnold conjectures (obtained in different cases by Arnold (1965), by Conley and Zehnder (1983), by Floer (1986) etc) leading to the creation of symplectic and contact topologies and to the recent proof of the necessity of the 4 cusps for every wave front eversion. (Obtained by Ju. Chekanov with P. Pushtar, 2000.)
The creation of contact and symplectic geometry and topology is one of the main discoveries at the boundary between mathematics and physics in the last half of the 20-th century (1965-2000). It is a new confirmation of the essential unity of both sciences.