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Thursday, 17th May 2001, 4:00 pm

Mathematics Bldg., lecture hall 2

Vladimir Arnold

(Steklov Institute, Moscow and Universite Paris-Dauphine)

"Wave propagation theory, Legendrian knots and contact topology"

** Abstract: **

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.

Coffee, Cookies at the faculty lounge at 3:30.

You are invited to join the speaker for further discussion after the talk at Beit Belgia.

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