  In this paper we study the effect of the Generalized Uncertainty Principle
(GUP) on the shadow of GUP-modified Kerr black hole and the correspondence
between the shadow radius and the real part of the quasinormal modes (QNMs). We
find that the shadow curvature radius of the GUP-modfied Kerr black hole is
bigger compared to the Kerr vacuum solution and increases linearly
monotonically with the increase of the GUP parameter. We then investigate the
characteristic points of intrinsic curvature of the shadow from a topological
point of view to calculate the the angular size for these curvature radii of
the shadow. To this end, we have used the EHT data for the M87* black hole to
constrain the upper limits of the GUP parameter red and our result is
$\beta<10^{95}$. Finally, we have explored the connection between the shadow
radius and the scalar/electromagnetic/gravitational QNMs. The GUP-modified Kerr
black hole is also used to provide perfect curve fitting of the particle
oscillation upper and lower frequencies to the observed frequencies for three
microquasars and to restrict the values of the correction parameter in the
metric of the modified black hole to very reasonable bound $\beta<10^{77}$.

General Relativity and Quantum Cosmology