I'll give an overview of the theorems we're going to spend the quarter proving, then do some organizing for coming weeks.

Computations in $RO(G)$-graded Bredon cohomology can be challenging and are not well understood, even for $G=C_2$ the cyclic group of order two. In this talk I will present a structure theorem for $RO(C_2)$-graded cohomology with coefficients in the constant $\mathbb{Z}/2$ Mackey functor that substantially simplifies computations. The structure theorem says the cohomology of any finite $C_2$-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will sketch the proof, which depends on a Toda bracket calculation, and give some examples.

Let $G$ be a finite group. In this talk, I will explain how the theory of $N_{\infty}$ algebras and operads appears, when viewed in the category of small $G$-categories. I will summarize the classification of $N_{\infty}$ operads, and also outline how this classification is reflected on the level of algebras. No prior knowledge of $N_{\infty}$ structures will be assumed.

For $G$ a compact Lie group and $X$ a finite $G$-CW complex, I will give a classification of the localizing tensor ideals of the derived category of cochains on the Borel construction of $X$, as well as the thick tensor ideals of the compact objects. The classification uses a support theory for tensor triangulated categories where the endomorphism ring of the unit is Noetherian, which I will assume no
knowledge of.

The Grassmannian manifold of $k$-planes in $\mathbb{R}^n$ has a $C_2$ action if $\mathbb{R}^n$ is taken to be a real representation of $C_2$. In this setting, the Schubert cell construction of the Grassmannian generalizes to an equivariant representation-cell structure. However, this generalization is not unique; an identification of the representation with $\mathbb{R}^n$ must be chosen. This non-canonicality proves to be advantageous. I’ll explain how we can discover the behavior of cellular spectral sequences for these spaces’ cohomology while remaining agnostic about their differentials and their structure at any finite stage.

Given a space with a $C_2$-action, we can consider the $RO(C_2)$-graded Bredon cohomology of the space. Generally computations in this theory are difficult, but using equivariant surgery, computations have now been done for all $C_2$-surfaces in coefficients given by both the constant $\underline{\mathbb{Z}}$ and constant $\underline{\mathbb{Z}/2}$ Mackey functors. In this talk I will describe the cohomology of $C_2$-surfaces in $\underline{\mathbb{Z}/2}$-coefficients. Furthermore, I will explain how we can represent all generators of the cohomology by fundamental classes of certain equivariant submanifolds. Time permitting, I will also discuss the slightly stranger results for $\underline{\mathbb{Z}}$-coefficients.