Georgia Institute of Technology College of Sciences

Recent work studying relations between moduli spaces in algebraic and tropical geometry reveals new and previously unexpected structures in the weight zero cohomology of moduli of curves and abelian varieties. Surprisingly, the motivic Lie algebra, a free Lie algebra with one generator in each odd degree greater than or equal to 3, whose representations control the category of mixed Tate motives over Z, appears in the weight zero cohomology of both of these spaces, with no natural map between the two. I will discuss these structures and highlight recent work of Amy Bradford that connects these two copies of the motivic Lie algebra, through the weight zero cohomology of the moduli of curves of compact type.