Grothendieck-Teichmüller theory and modular operads

The absolute Galois group of the rationals Gal(Q) is one of the most important concepts in number theory. Although we cannot explicitly describe more than two elements in this infinite group, we know it acts on well-known algebraic and topological objects in compatible ways. Grothendieck-Teichmüller theory uses these representations to study this Galois group. One of the most important representations comes from the compatible actions of Gal(Q) on all the profinite mapping class groups of surfaces. In this talk, we introduce an algebraic tool called an infinity modular operad and use it to construct an infinity modular operad of surfaces capturing the compatibility structure above. We show this admits an action of Gal(Q), translating the Grothendieck-Teichmüller program into the theory of infinity modular operads, which provides new ideas and tools to approach this problem. This is joint work with Marcy Robertson.