My main interest are concerned with all things spectral algebraic geometry. In particular I am interested in (equivariant) elliptic cohomology.
Spectral toric geometry: Inspired by Lurie's survey of elliptic cohomology I try to lift toric varieties to flat schemes over the sphere spectrum. These techniques should be used to lift the Tate curve and the universal generalized elliptic curve (leading to the moduli stack of oriented generalized elliptic curves).
Strict log geometry (with Elmo Vuorenmaa): Log geometry has been generalized to the ring spectrum setting by John Rognes. Meanwhile Mike Hill and Taylor Lawson used classical log geometry to define Tmf with level structures. It turns out that Rognes' generalization does not capture the the correct geometry to incorporate the geometry of Hill-Lawson. Therefore, we develop strict log geometry, a less general but more rigid context to capture the picture Hill and Lawson dictate.
Geometry of Tmf with Level structures: An alternative approach of Tmf with level structures in the spirit of Cestanvicius' "Moduli description of X_0(N)".
Equivariant Tmf (with Lennart Meier): Defining equivariant notions of Tmf and studying its properties compared to the work of Meier-Gepner and Lin-Tominaga-Yamashita.
Orientations of Thom spectra (with Leonard Tokic): We try to construct orientations of Thom spectra using equivariant techniques. In particular, we try to produce the trivializations needed to produce maps out of MUP (by work of Rok Gregoric) by understanding the line bundles \omega as sheaves obtained from equivariant TMF (as in Bauer-Meier).