Work

# Motivic Contractibility of the Space of Rational Maps.

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The moduli stack of bundles on a smooth complete curve over a field, is an immensely rich geometric object and is of central importance to the Geometric Langlands program. This thesis represents a contribution towards a motivic, in the sense of Voevodsky and Morel-Voevodsky, understanding of this stack. Following the strategy of Gaitsgory and Gaitsgory-Lurie we view the Beilinson-Drinfeld Grassmanian, $\\Gr_G(C)$ as a more tractable, homological approximation to $\\Bun_G(C).$ In the main theorem of this thesis we prove, using two different approaches, that the motive of the fiber of the approximation map is, in a number of different and precise ways, motivically contractible. This fiber is the space of rational maps, as introduced by Gaitsgory. One approach is to work in the context of modules where is a motivic $\\Ecal_{\\infty$-ring spectra and show that there is a motivic equivalence between the space of rational maps and a version of the $\\Ran$ space. Via various realization functors, we obtain the contractibility theorems of Gaitsgory and Gaitsgory-Lurie. A second, novel approach is to study the unstable motivic homotopy type using a theorem of Suslin and a model of the space of rational maps as introduced by Barlev.