Abstract: While some results are known for small g and n, making general
statements on the full cohomology ring H*(M_g,n) of the moduli space of
genus g, n-pointed smooth curves is very hard.  A key strategy to
approach this problem is to analyse a natural subring RH*(M_g,n), called
tautological ring, which is easier to work with but still contain a
large number of interesting algebraic cohomology classes. In the
seminar, we aim to address the natural question of whether the equality
RH*=H* occurs.
After an introduction on moduli spaces of curves and their tautological
rings, I will discuss joint work together with V. Arena, S. Canning, E.
Clader, R. Haburcak, A.Q. Li, and S.C. Mok and joint work with D. Faro
on the construction of many new non-tautological algebraic cohomology
classes arising from double cover cycles, generalising previous work by
Graber-Pandharipande and van Zelm.