First section was the Tate Construction for Finite Groups. One of the main arguments here was constructing the norm map $Nm_G: X_{hG} \to X^{hG}$. Where $X_{hG}$ is the homotopy orbits functor and $X^{hG}$ is the homotopy fixed points functor. The construction itself is outlined in I.1.7 of Nikolaus-Scholze.
The second section was on proving the Tate orbit lemma, which needed many preliminary results.
The third section was showing that the functor $-^{tG}: Sp^{BG} \to Sp$ admits a unique lax symmetric monoidal structure which makes the natural transformation $-^{hG} \to -^{tG}$ lax symmetric monoidal. The proof of this uses Verdier localizations of stable $\infty$-categories.
Started to read Chapter 5 in Bruno Martelli's Introduction to Geometric Topology:
This section is about the Sphere at Infinity
Started with Limit sets which are defined to be the accumulation(limit) points of the orbit $\Gamma(x)$. Proved two things:
(1) limit sets do not depend on $x$, with the crux of the proof relying on the fact that given two sequences $\{x_n\}, \{y_n\}$ with bounded $d_{\mathbb{H}^n}(x_n, y_n)$ and $x_n \to z$ then $y_n \to z$ as well.
(2)If $\Gamma'$ has finite index in $\Gamma$ then $\Lambda(\Gamma') = \Lambda(\Gamma)$. The main idea behind this proof is that given that we have finite index we can decompose $\Gamma$ into its cosets meaning that the subgroup orbits can be used to write the orbits for $\Gamma$, giving us the result.
Studied for Real Analysis qual: countability and separability axioms.