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In this talk, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $p$-adic $\text{GL}(2N, F)$ which we call \textit{middle supercuspidal representations}. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$. Furthermore, we pose a conjecture which refines the local converse theorem for general supercuspidal representations of $\text{GL}(n, F)$.

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A finite tensor category is a suitably nice abelian category with a compatible monoidal structure. It makes perfect sense to define the representation type of such a category, as a measure of how complicated the category is in terms of its indecomposable objects. For example, finite representation type means that the category contains only finitely many indecomposable objects, up to isomorphism.  

In this talk, we shall see that if a finite tensor category has finitely generated cohomology, and the Krull dimension of its cohomology ring is at least three, then the category is of wild representation type. This is a report on recent joint work with K. Erdmann, J. Plavnik, and S. Witherspoon. 

The question of profinite rigidity asks whether the isomorphism type of a group Γ can be recovered entirely from its finite quotients. In this talk, I will introduce the study of profinite rigidity in a different setting: the category of modules over a Noetherian domain Λ. I will explore properties of Λ-modules that can be detected in finite quotients and present two profinite rigidity theorems: one for free Λ-modules under a weak homological assumption on Λ, and another for all Λ-modules in the case when Λ is a Dedekind domain. Returning to groups, I will explain how these algebraic results yield new answers to profinite rigidity for certain classes of solvable groups. Time permitting, I will conclude with a sketch of future directions and ongoing collaborations that push these ideas further.