Fields Colloquium

Moduli spaces are important in both mathematics and theoretical physics. In this talk, I will introduce the basic ideas behind their construction and explain how recent discoveries linking the moduli space of curves to the moduli space of graphs have led to new insights into their topology. Interestingly, the cohomology of these spaces is very high-dimensional, meaning there is a kind of "dark matter" present—much structure exists, but little is explicitly known about it. One especially interesting aspect of this work is that it reveals just how intricate the topology of these spaces can be. I will also explain how techniques from physics and quantum field theory have emerged as surprisingly effective tools for computations in this branch of mathematics.