The modified Adams--Novikov spectral sequence and a deformation of Borel equivariant homotopy

The R-motivic stable homotopy category has a close connection to the C2-equivariant category via the C2-equivariant Betti realization map. For the bigraded homotopy groups, the C2-equivariant spectra are usually more complicated to compute than the R-motivic ones, due to the existence of the "negative cone”. I will first explain this connection; then I will talk about the joint work with Gabriel Angelini-Knoll, Mark Beherens, and Eva Belmont, in which we construct the modified Adams--Novikov spectral sequence (mANSS) aiming to build an odd primary analog of this R-motivic and C2-equivariant relation. Using the mANSS filtration, one can construct a deformation of Borel equivariant homotopy, which agrees with the a-completed Artin—Tate R-motivic category for group C2.