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Anubhav Nanavaty : Weight Filtrations and Derived Motivic Measures (Nov 18, 2024 3:10 PM)

Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher K groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of K_0) to so-called "derived" ones, i.e. on the level of K_i for all i. I will describe some of this work, which shows that if one closely studies how the Gillet-Soulé weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the motivic stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher K groups of varieties, and relating them to other interesting algebro-geometric objects in the literature.

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