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The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy… (More)

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent… (More)

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the… (More)

We study the singular homology (with field coefficients) of t he moduli stack Mg,n of stablen-pointed complex curves of genus g. Each irreducible boundary component ofMg,n determines via the… (More)

The Nielsen Realization problem asks when the group homomo rphism Diff(M) → π0Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but… (More)

Given an integral scheme X over a non-archimedean valued field k, we construct a universal closed embedding of X into a k-scheme equipped with a model over the field with one element F1 (a… (More)

We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of… (More)

We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image — it fails to be essentially surjective in a very… (More)

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra… (More)

Using certain Thom spectra appearing in the study of cobord ism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler… (More)