Let G be a reductive group, E an elliptic curve, and Bun_G the moduli stack of principal G-bundles on E. In this talk, I will attempt to explain why Bun_G is a very interesting object from the perspectives of both singularity theory on the one hand, and shifted symplectic geometry and representation theory on the other. In the first part of the talk, I will explain how to construct slices of Bun_G through points corresponding to unstable bundles, and how these are linked to certain singular algebraic surfaces and their deformations in the case of a "subregular" bundle. In the second (probably much shorter) part, I will discuss the shifted symplectic geometry of Bun_G and its slices. If time permits, I will sketch how (conjectural) quantisations of these structures should be related to some well known algebras of an "elliptic" flavour, such as Sklyanin and Feigin-Odesskii algebras, and elliptic quantum groups.
