We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral sequences that compute homotopical data starting with purely algebraic data.

We compute the bigraded homotopy ring of the Borel C 2 -equivariant K(1)local sphere. This captures many of the patterns seen among Im J-type elements in R-motivic and C 2 -equivariant stable stems. In addition, it provides a streamlined approach to understanding the K(1)-localizations of stunted projective spaces.

We introduce the F -motivic lambda algebra for any field F of characteristic not equal to 2. This is an explicit differential graded algebra whose homology is the E 2 -page of the F -motivic Adams spectral sequence. Using the R-motivic lambda algebra, we compute the cohomology of the R-motivic Steenrod algebra through filtration 3. This is the algebraically universal case, yielding information about the cohomology of the F -motivic Steenrod algebra for any base field F .We then study the 1-line of the F -motivic Adams spectral sequence in detail. In particular, we produce differentials d 2 (h a+1 ) = (h 0 + ρh 1 )h 2 a valid over any base field F , as well as the following computations in the F -motivic Adams spectral sequence for particular base fields F . For F of the form R, Fq with q an odd prime-power, Qp with p any prime, or Q, we determine the 1-line of the E 3 -page of the F -motivic Adams spectral sequence, as well as all higher differentials in stems s ≤ 7; for F = R, we determine all permanent cycles on the 1-line; and for F = Fq or F = Qp with q, p ≡ 1 (mod 4), we determine all differentials out of the 1-line.These computations are stable motivic analogues of the classic Hopf invariant one problem. We also consider the unstable motivic analogue, showing that it reduces to known results with a finite number of exceptions. As an application, we classify which unstable motivic spheres may be represented by smooth schemes admitting a unital product. * , * is a fundamental computational tool in motivic stable homotopy theory. Here, A F is the F -motivic Steenrod algebra, which acts on M F , the mod 2 motivic cohomology of Spec(F ). This spectral sequence converges to π F * , * , the homotopy groups of the (2, η)-completed F -motivic sphere. This spectral sequence has been used frequently over the past decade. Dugger, Isaksen, Wang, and Xu [DI10, Isa19, IWX20a] have made deep computations for F = C, leading to the current best results on classical stable stems [IWX20b]. Belmont, Dugger, Guillou, and Isaksen [DI16a, DI17, GI20, BI20b, BGI21] made extensive computations for F = R, with applications to C 2 -equivariant stable stems. Wilson and Østvaer [Wil16,WØ17] analyzed the spectral sequence for F = F q to study motivic stable stems over finite fields in weight zero.

We compute the RO(A)-graded coefficients of A-equivariant complex and real topological K-theory for A a finite elementary abelian 2-group, together with all products, transfers, restrictions, power operations, and Adams operations.

We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding wellbehaved theories of power operations for E∞ ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction-theoretic machinery, we obtain tools for computing with E∞ algebras over Fp and over Lubin-Tate spectra. As an application, we demonstrate the existence of E∞ periodic complex orientations at heights h ≤ 2.