My research so far has been concentrated mainly on equivariant homotopy theory and algebraic K-theory. Recently I have been thinking of a way of using the tools of those two disciplines to better understand the stable motivic category, both via the construction of a motivic Hermitian K-theory spectrum and via a detailed study of the family of Betti realizations.
Hermitian K-theory for stable ∞-categories IV: Poincaré motives and Karoubi-Grothendieck-Witt groups
with Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Kristian Moi, Markus Land, Thomas Nikolaus and Wolfgang Steimle. In preparation
Crossing Browder's bridge
Forthcoming. An explanation in modern language of Browder's original proof of his theorem relating the Kervaire invariant of framed manifolds and the 2-line of the Adams spectral sequence.