Topology and combinatorics of low genus mapping spaces to projective space
This thesis studies the topological invariants and boundary combinatorics of two classes of moduli spaces parametrising maps from elliptic curves to projective space - the Vakil--Zinger space and Kontsevich stable map spaces - combining the techniques of mixed Hodge structures, tropical moduli spaces, symmetric functions, and the Grothendieck ring of varieties. The results of the thesis extend the structures and results on the moduli spaces of low genus stable curves and genus zero stable maps to projective space, overcoming the subtlety caused by smoothability, and point the way towards higher genus generalisations.
Moduli space, Boundary complex, Symmetric function, Mixed Hodge structure
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