Derived analytic geometry and invariants

• Applicant: Dr. Devarshi Mukherjee
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 534946574

This proposal investigates several interactions between (derived) analytic geometry and noncommutative geometry using newly developed approaches to homotopy theory in functional analytic settings - bornologies and condensed mathematics. Classically, analytic and algebraic geometry have been used to study fundamental geometric objects such as varieties, analytic spaces and manifolds, which are locally modeled by commutative topological rings. It turns out, however, that there are several natural geometric phenomena such as the formation of quotients of group actions on spaces, non-transversal intersections of curves, that necessitate working with algebraic objects that capture the additional complexity. These objects are derived commutative (topological) rings, which are the building blocks of derived (analytic) geometry. Another way to model such geometric phenomena is through noncommutative topological algebras, and invariants of such algebras, which lie at the core of noncommutative geometry. Remarkably, noncommutative invariants such as topological cyclic homology and algebraic K-theory have also proved to be very powerful tools in the study of purely algebraic and commutative objects such as varieties and derived schemes. In this project, we propose an extension of these invariants (namely, topological cyclic homology) and the techniques of their use to analytic spaces, and more generally, derived analytic spaces. Such constructions will be carried out using the recently developed frameworks of bornological and condensed mathematics, which are ideal foundations for the interactions between analysis and (homotopical) algebra. We will also use techniques from derived analytic geometry to compute cyclic homology for a certain nonarchimedean completion of a group ring for groups that are of significance in geometric group theory and the celebrated Langlands program.

• DFG Programme: Research Grants