K-theory · Algebraic K-theory -- Arithmetic and Topology
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-10-01 → 2022-09-30
- EU contribution
- €219,312
- Participants
- 1
- Scheme
- MSCA-IF
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Results in brief
Algebraic K-theory -- Arithmetic and Topology
Algebraic K-theory was invented by Grothendieck in his formulation of the Riemann--Roch theorem in algebraic geometry. Since then, the ideas involved in its definition have lead to numerous related invariants such as topological K-theory, higher algebraic K-theory, Milnor K-theory, and hermitian K-theory. These invariants play a role in many areas of mathematics: operator algebras, homotopy theory, and algebraic and arithmetic geometry. As such, both the understanding of the structural behaviour, as well as explicit computations of these invariants have seen many breakthroughs in the area over decades of mathematical research. The purpose of the EU funded project ``Algebraic K-theoy: Arithmetic and Topology'' (K-theory) was to build on recent advances in algebraic and hermitian K-theory and to study both structural and computational results, relating K-theory to questions in arithmetic and algebraic geometry on the one hand and to topology on the other hand. The objectives in the project were three-fold: - The first objective was to find a formula for a ring spectrum, called the circle-dot ring, the ER had found in earlier work with Georg Tamme, which is more amenable for computations that its definition. Having such a formula, together with earlier work of Tamme and the ER, gives new methods for computations of K-theories of classical objects. - Generally, algebraic K-theory is a complicated object to compute. However, the situation becomes more tractable when one simplifies the algebraic K-theory, for instance by neglecting torsion, or considering only p-primary information for a prime number p. Chromatic homotopy theory is a systematic study of various (more drastic) ways of simplifying objects such as algebraic K-theory. The purpose of the second objective was to study chromatic simplifications of algebraic K-theory both from structural and computational points of views. - The last objective was about relating recently developed fibre sequences in the theory of hermitian K-theory to know long exact sequences describing bordism theories of Poincar\'e duality complexes (rather than manifolds): Both of the sequences contain a common term, and the purpose was to find a general explanation for this common term to appear.
Data: CORDIS, © European Union
Project objective
Algebraic $K$-theory -- Arithmetic and Topology. The goal of the proposed project is to use recent results in algebraic K-theory to further the connection between algebraic K-theory on the one hand, and arithmetic and topology on the other. This will split the proposed project into two parts: The one exhibiting connections to topology, more precisely the topology of manifolds and Poincaré duality spaces, and the other exhibiting connections to arithmetic geometry, more precisely gaining access at explicit calculations of K-theory groups of (spectral) schemes. Both main goals build on previous work in which I played a role, on the one hand establishing the foundations of a new ``real algebraic K-theory spectrum'' which leads in particular to a solution of a conjecture of Hesselholt--Madsen, and on the other hand a recent result in algebraic K-theory which proves the existence of a ring spectrum, the circle-dot ring, which determines the failure of excision in K-theory. The pure existence (and some formal properties) of this ring have already been exploited for many applications, and the goal of this part of the project is to make the circle-dot ring more explicit and use this new knowledge for explicit computations.
Original text from CORDIS.
Participants
- KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark
Links
Data: CORDIS, © European Union