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We report significant progress toward establishing an obstruction theory of equivariant Thom spectra with multiplicative structures arising from maps of $E_V$ spaces. Focusing on the Fujii-Landweber Real bordism spectrum, we explain an argument to show that a homotopy ring Real orientation of an $E_{\rho}$-spectrum satisfying strong equivariant evenness and mild additional commutativity condition lifts to an $E_{\rho}$ map.First, we address the foundational issues. We discuss model categories for equivariant $C_2$-spectra indexed on Real inner product spaces and comparisons among them. We explain which foundations are needed for our project and describe the parts that would be original even non-equivariantly.Next, we construct equivalences between several $E_V$ operads, including little disks and Steiner operads. We show that algebras over $E_{V \oplus \Rbb}$ operads can be strictified to monoids in the category of $E_V$-algebras.To compute obstruction groups for maps of $E_V$ algebras, we analyze the spectra of the derived indecomposibles of augmented $E_V$-algebras. We determine that such spectra are $V$-fold desuspensions of a lift of the May delooping machine to the augmented algebras.Next, we discuss the induced map on cohomology for a map between spaces whose $RO(C_2)$-graded cohomology carries obstruction classes for homotopy ring and $E_\rho$ ring maps out of the Real bordism spectrum.Finally, we review the strategies for the proofs of the goal results of our project, explaining how to use the technique of climbing the slice tower of the target to construct highly structured orientations out of Real bordism.

Equivariant algebra is the study of algebra in the category of Mackey functors. In this setting, Mackey functors play the role of abelian groups and incomplete Tambara functors play the role of commutative rings. The study of incomplete Tambara functors parallels the study of classical commutative algebra in many ways, but there are some striking differences as well. We develop aspects of the homological and commutative algebra of incomplete Tambara functors in this thesis. One of the most notable differences is the fact that free incomplete Tambara functors often fail to be free as Mackey functors, even though free algebras (polynomial rings) are always free as modules. We provide conditions under which a free incomplete Tambara functor is flat as a Mackey functor. When the group is solvable, we show that a free incomplete Tambara functor is flat precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.

We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. We introduce the notion of geometrically oriented spectra, which extends the notion of complex oriented spectra, and of which the geometric cobordism spectrum is the universal example. Other examples of geometrically oriented spectra include the Eilenberg-Maclane spectrum associated to a constant Mackey functor, and the connective cover of equivariant complex K theory. On the algebraic side, we define and study filtered equivariant formal group laws, which are the algebraic structures determined by geometrically oriented spectra. We prove some of the fundamental properties of filtered equivariant formal group laws, as well as a universality statement for the filtered equivariant formal group law determined by the geometric complex cobordism spectrum.

We study two invariants of topological Hochschild homology coming from equivariant homotopy theory: its RO(C [subscript p superscript n])-graded homotopy Mackey functors, and the regular slice filtration. In the case of RO(C [subscript p superscript n])-graded homotopy, we explain how to relate Angeltveit-Gerhardt's work to the gold elements, and in cases of interest give canonical identifications of the relevant groups in terms of the kernels of the Fontaine maps θ̃ [subscript r]. This is then used as input for studying the slice filtration on THH. When R is a torsionfree perfectoid ring, we show that the C [subscript p] -regular slice spectral sequence of THH(R; Z [subscript p]) collapses at E².