Harvard-MIT Combinatorics: Configuration spaces on graphs, phylogenetic trees, and moduli spaces of tropical curves


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November 16, 2022 4:15 pm - 5:15 pm
Science Center 232

Melody Chan - Brown University

I will discuss joint work with Christin Bibby, Nir Gadish, and Claudia Yun.  The historical antecedents are in earlier work of Billera-Holmes-Vogtmann around 2000 and others, who study an interesting space of metric trees on n labelled taxa. This is a space that is shellable and whose top homology was calculated, as an S_n-representation, by Robinson-Whitehouse.  These spaces have a reinterpretation as moduli of tropical curves of genus 0.  Other historical antecedents of our work are in the study of configuration spaces of n points on a graph, whose topological invariants are quite interesting. For example, Ko-Park proved that failure of planarity of a graph can be detected by torsion in H_1 of its unordered configuration space.

The work I will then describe concerns a genus g>0 analogue of the space of phylogenetic trees: the moduli space of tropical curves of genus g. Roughly speaking, this space parametrizes n-marked graphs of first Betti number g.  These spaces are no longer shellable for g>1, and their homology groups, as S_n-representations, are quite mysterious.  I will explain how making precise connections to compactified configuration spaces on graphs made it possible for us to make calculations in Sage when g=2 in a range beyond what was previously accessible.  These in turn produced new calculations and conjectures concerning the rational cohomology of M_{2,n}. No tropical geometry prerequisites are assumed in this talk.