Arnold's Seminar

  March 23, 1999

P.Pushkar',  Lagrangian intersections in a symplectic vector space.

Summary.

I would like to speak about the following theorem.

Theorem.
Let L be the product of the standard unit circles lying
in the coordinate symplectic planes of the standard
symplectic space R^4. Then for a generic symplectic
linear operator A the image A(L) meets L at least in 8 point.
The estimate is sharp, and one can construct an example with
exactly 8 points.

This theorem answers a question due to L.Lerman.

We will also construct a Lagrangian submanifold L^n in R^{2n}
(which coincides with the n-dimensional torus only for n=1,2!)
and prove a similar rezult:  for generic A the image A(L^n)
meets L^n at least in 4n points.