Title pretalk: compactifications of $M_{1,n}$ with Gorenstein curves.
Abstract pretalk: I will describe the geometry of isolated Gorenstein curve singularities of genus one, and how they can be used to build a multitude of alternative compactifications of $M_{1,n}$.
Title: Chow rings of moduli spaces of curves of genus one with few markings.
Abstract: Modular compactifications of $M_{1,n}$ parametrising only Gorenstein curves have been constructed and classified by Smyth and Bozlee-Kuo-Neff (there are a lot of them). The combinatorially simplest one $U_{1,n}$ parametrises curves without rational tails; for $n \leq 6$, Lekili and Polishchuk identified it with a weighted projective stack or a Grassmannian. In joint work with Andrea Di Lorenzo, we consider the Artin stack $G_{1,n}$ of log- canonically polarised Gorenstein curves of genus one. It admits a stratification by tail type, whose strata are products of $U_{1,m}, m \leq n$ with moduli spaces of stable rational curves. For $n \leq 6$, we find an explicit description of its integral Chow ring by patching. The Chow ring of any modular compactification (including $\overline{M}_{1,n}$) can be obtained from $A^*(G_{1,n})$ by excision. Moreover, these spaces satisfy the Chow-Künneth generation property (implying rational Chow=cohomology and polynomial point cunt).