Delocalized Equivariant Elliptic Hochschild Homology

In this thesis we introduce the notion of \emph{Elliptic Hochschild Homology} of derived stacks in characteristic zero. This notion is studied and some fundamental properties are shown, and it is computed in simple cases. We then introduce its \emph{periodic cyclic} version and prove it recovers Grojnowski's equivariant elliptic cohomology of the analytification for quotient stacks. In the second part of the thesis, we provide a notion of $k$-rationalized equivariant elliptic cohomology for $\bQ$-algebras $k$, via adelic descent. We study the adelic decomposition of equivariant cohomology and K-theory and prove comparison theorems with periodic cyclic homology variants of the theories. Finally, we collect partial results and ideas that will be explored in future work.