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Explore a detailed mathematical seminar presentation that delves into recent findings concerning the universal localizing invariant of stable categories over base ring k and its relationship with filtered colimits. Learn about the category of localizing motives (Mot^loc_k) and its properties as a symmetric monoidal category, including its rigidity in the Gaitsgory-Rozenblyum sense. Discover how Mot^loc_k can be conceptualized as a category of quasi-coherent sheaves on a specialized stack, and understand the computational methods for morphisms within this category. Examine the corepresentability results for topological cyclic homology (TC) and TR when applied to connective E_1-rings. Gain insights into the refined negative cyclic homology and its variants, particularly focusing on cases where the base E_∞-ring k is Z-linear or complex oriented, and understand its relationship with the category Nuc(k[[u]]).
