Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.