**Jong Hae Keum**

Korea Institute for Advanced Study, South Korea

January 15, 2014

**K3 Surfaces: Automorphisms, Algebra and Geometry**:
A review on $K3$ surfaces will be given, their history and recent result on them.
It is a natural and fundamental problem to determine all possible
orders of automorphisms of $K3$ surfaces in any characteristic. Even
in the case of complex $K3$ surfaces, this problem has been settled
only for symplectic automorphisms and purely non-symplectic
automorphisms, due to Nikulin, Mukai, Kondo and Oguiso.
In a recent work I solve the problem in all characteristics bigger than $3$. In particular, $66$ is the maximum possible finite order in each characteristic bigger than $3$.
I will also give an interesting comparison with the corresponding well
known result on elliptic curves.