Pim Spelier: The log tautological ring of the moduli space of curves
Abstract: The tautological ring of Mgn-bar has been a crucial object in the study of the intersection theory of the moduli space of curves. Recently, there has been more focus on the logarithmic enumerative geometry of Mgn-bar, with interesting classes coming from e.g. log double ramification cycles. We present a definition of the log tautological ring of Mgn-bar, together with a log decorated strata algebra, and prove several structure results. The main new tools are the notions of cone stacks with boundary and homological piecewise polynomials, that capture the tropicalisation of strata of log smooth stacks and the combinatorial part of their intersection theory.
This is joint work with Rahul Pandharipande, Dhruv Ranganathan and Johannes Schmitt.

Lucas Gierczak: Counting Weierstrass points on degenerating algebraic curves
Abstract: Weierstrass points on algebraic curves are special points of high importance in algebraic geometry and arithmetic geometry. In this talk, we study how those special points behave when the algebraic curve degenerates to a nodal curve. To this end, we first explain why tropical geometry is a relevant formalism for studying degeneration questions. We then define a tropical analogue on metric graphs (seen as tropical curves) for Weierstrass points, and explore the properties of the so-called “tropical Weierstrass locus". We also associate intrinsic weights to the connected components of this locus, and show that their total sum for a given metric graph and divisor is a function of few combinatorial parameters (degree and rank of the divisor, genus of the metric graph). Finally, in the case the divisor on the metric graph comes from the tropicalization of a divisor on an algebraic curve, we specify the compatibility between the Weierstrass loci.

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