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In many ways the state of our understanding of homeomorphisms of 4-manifolds at the end of 2023 is essentially that of our understanding of homeomorphisms of 2-manifolds in 1973, before Thurston changed everything. In two talks I will report on some first steps in a project (joint with Eduard Looijenga) whose ultimate goal is to address this. I will focus on the case of K3 surfaces.  One appealing aspect of this topic is the surprising variety of concepts that naturally arise, including : the theory infinite reflection groups; Hodge theory; the theory of integral quadratic forms;  period mappings; families of elliptic curves;  and much more.  

In this first talk I'll start with the big picture, and then explain how an idea from number theory can be used to understand mapping class groups of K3 surfaces.