Algebraic K-theory via dualizable categories and motivic filtrations

• Applicant: Dr. Hyungseop Kim
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 576038862

This proposal investigates properties and constructions related to algebraic K-theory, localizing invariants and their motivic filtrations, with a particular emphasis on applications to K-theory of ring spectra, analytic spaces and stacks, using recent techniques from dualizable categories. Within the theme of six-functor formalisms, I propose to (1) develop a six-functor formalism for noncommutative motives over rigid bases and study its applications to compactly supported localizing invariants and Poincaré-type duality for ring spectra, and (2) prove A^1-invariance of continuous K-theory on regular adic spaces as a foundational question in the emerging motivic homotopy theory of analytic spaces. As applications and a continuation of formal gluing and descent for localizing invariants on dualizable categories studied in my earlier work, I intend to (3) describe formal gluing for manifolds, study the K-theory of Morava E-theory via such gluing, and establish a continuous adelic descent statement for localizing invariants including suitable nuclear module categories for archimedean parts. For stacks, I aim to (4) compute the topological cyclic homology of syntomic stacks as an approximation to the K-theory of motives and develop relevant tools to approach K-theory of stacks in general. Finally, toward motivic filtrations, I will work on (5) steps toward constructing such a filtration on continuous K-theory of analytic spaces, and (6) construct motivic filtrations on K(n)- or T(n)-local topological cyclic homology for appropriate ring spectra, with the aim of linking the purity equivalence to certain cohomology theories.

• DFG Programme: Position