What's the problem with taking quotients?

Title: What's the problem with taking quotients? 

Abstract: As we have noted earlier in this seminar, taking quotients can be quite painstaking. There is no guarantee that the naive functor G/H gives a group scheme, and there is even no information on whether the universal property guarantees the existence of group schemes. What do we need to make this work? In this talk, we are going to briefly include Grothendieck topologies and conditions for making quotients work nicely, so that we can put our hearts back to our chest.