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Motivated by extremal isoperimetric problems in percolation, I shall describe a

model which puts a global curvature constraint on the classical 
Ulam's problem in the plane, and studies the longest increasing path 
from (0,0) to (n,n) trapping atypically large area. As is typical in these 
models, the first order behaviour of this random contour is determined by a 
variational problem which we explicitly solve. More interesting are exponents 
related to local fluctuation properties which capture the competition between the 
global curvature constraint and the behaviour of an unconstrained path governed 
by KPZ universality. These can be studied via maximal facet lengths of the convex 
hull of the contour and the Hausdorff distance from the hull for which we identify 
scaling exponents 3/4 and 1/2 respectively.  I shall also discuss  connections 
to different models  and several open problems.

Joint work with Shirshendu Ganguly and Alan Hammond.