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Abstract
This dissertation studies the representation theory of classical Lie superalgebras from a categorical point of view. Given a classical Lie superalgebra $\fg = \fg_{\bar 0} \oplus \fg_{\bar 1}$, one can consider the category $\mathcal{C}_{(\fg, \fg_{\bar 0})}$ of $\fg$-supermodules which are semisimple as modules over the Lie algebra $\fg_{\bar 0}$. Also, one can consider the full subcategory $\mathcal{F}_{(\fg, \fg_{\bar 0})}$ of $\mathcal{C}_{(\fg, \fg_{\bar 0})}$ which consists of the finite-dimensional supermodules. These are Frobenius categories, so one can form the stable categories $\Stab(\mathcal{C}_{(\fg, \fg_{\bar 0})})$ and $\stab(\mathcal{F}_{(\fg, \fg_{\bar 0})})$ which are triangulated categories. The tensor product of supermodules gives $\Stab(\mathcal{C}_{(\fg, \fg_{\bar 0})})$ and $\stab(\mathcal{F}_{(\fg, \fg_{\bar 0})})$ the structure of tensor triangulated categories, which raises many deep questions about the tensor structure. \par Balmer associates to each essentially small tensor triangulated category $\mathscr{K}^c$ two topological spaces: $\Spc(\mathscr{K}^c)$ and $\Spc^{\text{h}}(\mathscr{K}^c)$, called the categorical (Balmer) spectrum and the homological spectrum respectively. When $\mathscr{K}^c$ is rigid, in all known examples the comparison map
$$
\phi: \Spc^{\text{h}}(\mathscr{K}^c) \to \Spc(\mathscr{K}^c)
$$
is a bijection. Balmer's Nerves-of-Steel Conjecture states that this is always the case. We prove the conjecture holds for $\mathscr{K}^c = \stab(\mathcal{F}_{(\fg, \fg_{\bar 0})})$ for $\fg$ a Type A Lie superalgebra. The argument involves using the detecting subalgebras introduced by Boe, Kujawa, and Nakano, as well as the stratification framework developed by Benson, Iyengar, and Krause. As a consequence, we are able to use the more recent h-stratification introduced by Barthel, Heard, Sanders, and Zou to classify localizing subcategories of $\Stab(\mathcal{C}_{(\fg, \fg_{\bar 0})})$, again for Type A Lie superalgebras.