Quebec Mathematical Sciences Colloquium

November 14, 2008 from 16:00 to 18:00 (Montreal/EST time) On location

Overcrowding and Undercrowding of Random Zeros on Complex Manifolds

Colloquium presented by Bernard Shiffman (Johns Hopkins University)

We discuss the distribution of zeros of random polynomials in m complex variables and, more generally, of random holomorphic sections of ample line bundles over compact Kaehler manifolds. We show that the zeros are highly likely to be uniformly distributed if the degree of the polynomial or line bundle is large. For example, the overcrowding and undercrowding probabilities for the volume of the zero set in a fixed domain decay like exp(-CN^{m+1}) as the degree N increases. We use off-diagonal asymptotics of the Bergman-Szego kernels to analyze coherent states centered at lattice points and to obtain large deviation estimates for the maximum modulus, and then we apply methods from Nevanlinna theory to obtain estimates for the zeros. This talk involves joint work with Steve Zelditch and Scott Zrebiec.


UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., room SH-3420