Euler characteristics of the universal Jacobian and mapping spaces

I will explain how to express the weight-graded compactly-supported Euler characteristics of the universal Jacobian in terms of those of the moduli space of curves. This leads to closed formulas for the weight-zero and topological Euler characteristics of the universal Jacobian, following earlier work of Chan—Faber—Galatius—Payne and Gorsky on the moduli space of curves. I will also discuss closely related joint work with Terry Song, in which we study the weight-graded Euler characteristics of the moduli space of maps from smooth pointed genus-g curves to projective space.