Equivariant Elliptic Cohomology

The subject of equivariant elliptic cohomology finds itself at the interface of topology, string theory, affine representation theory, singularity theory and integrable systems. These connections were already known to the founders of the discipline, Grojnowski, Segal, Hopkins, Devoto, but have come into sharper focus in recent years with a number of remarkable developments happening simultaneously: First, there are the geometric constructions, due to Kitchloo, Rezk and Spong, Berwick-Evans and Tripathy, building on the older program of Stolz-Teichner, and of course Segal. These are now all known to be equivalent to Grojnowski's original definition, thanks to work of Spong. Second, there are new applications, to integrable systems (Aganagic-Okounkov, Felder-Rimanyi-Varchenko-Tarasov) and to representation theory (Yang-Zhao-Zhong, G-Ram). Finally, there is the work of Weber-Rimanyi-Kumar, who build on the techniques of Borisov and Libgober to define Schubert classes in equivariant elliptic cohomology (with an added dynamical parametre), recovering the stable envelopes in type A. This talk will give a gentle introduction to the topic and attempt an overview over these recent developments.