# Random Functions, Random Surfaces and Interfaces

(Web site soon online)

January 4 - 9, 2009

Organizers: R. Bond (Toronto), M. Douglas (Rutgers), S. Shlosman (CNRS), S. Sheffield (New York), S. Zelditch (Johns Hopkins)

This workshop is devoted to random fields such as Gaussian random fields *f ~ ∑ _{1≤i≤∞} a_{i}(*w

*)*

*j*

*where {*

_{i}(x)*j*

*} is an orthonormal basis for a Hilbert space*

_{i}*H*and where the coefficients

*a*w

_{i}(*)*are independent (real or complex) Gaussian random variables of mean zero and variance one. Motivated by such physical models as

*(i)*the large scale matter distribution in the universe or

*(ii)*landscape statistics in string theory or

*(iii)*the random wave model in quantum chaos or

*(iv)*limit shapes of phase interfaces in statistical mechanics, the workshop will largely focus on the zeros or critical points of random fields.

Specific topics include: the Nazarov-Sodin theorem that the average number of nodal domains of random spherical harmonics of degree *N* has an asymptotic formula of a kind predicted by Bogomolny-Schmidt, Smilanksy and others; the Sheffield-Schramm results on the zero sets of the two-dimensional Gaussian free field and their relation to SLE_{6} curves; work of Douglas and collaborators Ashkok, Denef, Shiffman, Zelditch and others on the applications of random complex geometry to counting vacua in string-*M* models; results of Shlosman, Schonmann and others on properties of phase interfaces such as flatness of crystal facets.