In this project, joint with Robert Burklund, we study the theory of perfect $E_{\infty}$ algebra over field of characteristic $p$, as an application we recover the homotopy type of genuine fixed points via the Borel equivariant data for a finite $G$-complex in the spirit of the Sullivan conjecture. (Parts of the work is available in my thesis). 

This is the second part of the joint work with Burklund, in which I prove that the Segal conjecture for large objects also holds for arbitrary p-groups and its higher semiadditivity implications. (Parts of the work is available in my thesis)

We analyse the p-local type of homotopy fixed points of the loop rotation on the loop group using methods adjacent to the proof of the Sullivan conjecture, and compare it to the genuine fixed points described by G. Williamson and S. Riche. (Parts of the work is available in my thesis)

In joint work with Oscar Harr, we study a version of Smith category of equivariant sheaves in the directions of generalising the results due to David Truemann. As a preparation for we also write down a coherent Six functor formalism for G-equivariant sheaves for G, a locally compact Hausdorff group. (Parts of the work is available in my thesis).

In this project, I give a different proof the Sullivans' theorem which identifies the homotopy automprhisms of fintie rational space as rational points of an algebraic group. The methods involve use the Artin Representability theorem of Lurie.