  We study polymers attached to spherical (circular) or paraboloidal
(parabolic) probes in three (two) dimensions. Both self-avoiding and random
walks are examined numerically. The behavior of a polymer of size $R_0$
attached to the tip of a probe with radius of curvature $R$, differs
qualitatively for large and small values of the ratio $s=R_0/R$. We demonstrate
that the scaled compliance (inverse force constant) $S/R_0^2$, and scaled mean
position of the polymer end-point $<x_\perp>/R$ can be expressed as a function
of $s$. Scaled compliance is anisotropic, and quite large in the direction
parallel to the surface when $R_0\sim R$. The exponent $\gamma$, characterizing
the number of polymer configurations, crosses over from a value of $\gamma_1$ -
characteristic of a planar boundary - at small $s$ to one reflecting the
overall shape of the probe at large $s$. For a spherical probe the crossover is
to an unencumbered polymer, while for a parabolic probe we cannot rule out a
new exponent.
