Mathematicians are building a new kind of geometry that treats algebraic and analytic spaces as two sides of the same coin, using tools from homotopy theory to unify them. The problem is that algebraic K-theory—a central invariant for studying these geometric objects—is notoriously hard to compute because it lacks a property called excision. This forces researchers to rely on approximations like cyclic homology, which work well in some settings but fail in others. The project tackles two specific gaps: extending excision results from algebra to analytic geometry, and fixing periodic cyclic homology’s breakdown in positive characteristic. This is fundamental mathematics, not applied research. There is no immediate practical application. However, the tools developed here—continuous K-theory for bornological algebras and analytic cyclic homology—could eventually underpin advances in areas that rely on noncommutative geometry, such as quantum field theory or number theory. Past work in this vein has unexpectedly fed into cryptography and coding theory. A deeper understanding of these invariants may one day sharpen the mathematics behind secure communications or materials modelling, but that remains decades away.
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Recent work on the foundations of functional analysis has enabled the use of tools from homotopy theory to develop a unified approach towards derived and noncommutative geometry, expressing (derived) algebraic and analytic geometry as special cases. Derived geometry studies schemes and algebraic stacks in the algebraic case, and analytic spaces and stacks over a base Banach ring in the analytic setting. The way these geometric objects are often studied is through invariants of noncommutative algebras or categories of such algebras) that are naturally associated to them. The most fundamental such noncommutative invariant is algebraic K-theory, which is the so-called universal localising invariant. Algebraic K-theory is however hard to compute as it lacks an important property called excision, and can therefore only be approximated by cyclic homology and its variants. In this project, we will study K-theory and its approximations in the context of analytic and noncommutative geometry - both of which can be studied using the framework of bornologies, following the work of Ben-Bassat, Kremnizer and Meyer. More concretely, we will use the recently discovered continuous K-theory by Efimov to define a version of infinitesimal K-theory for bornological algebras and show that it measures the obstruction to excision in algebraic K-theory and negative cyclic homology, thereby extending a landmark result to the analytic setting. Opposite to K-theory is periodic cyclic homology, which satisfies several desirable properties when one works over a base field of characteristic zero. In positive characteristic, however, periodic cyclic homology is badly behaved. A recently developed invariant called analytic cyclic homology corrects its defects. In the second part of the proposal, we will show that this theory coincides with periodic cyclic homology for suitable rigid analytic spaces and nonarchimedean completed group algebras for hyperbolic and reductive p-adic groups.
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