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Cohomology and the transfinite

Outstanding problems in a variety of mathematical fields have been shown to reduce, in essence, to questions about the combinatorial structure of the transfinite; this was the case, for example, in Shelah's solution to the Whitehe... ver más

28/02/2026

181K€

Presupuesto del proyecto: 181K€

Líder del proyecto

UNIVERSIDAD DE BARCELONA No se ha especificado una descripción o un objeto social para esta compañía.

Total investigadores 327

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2023-03-21

HORIZON-MSCA-2022-PF-01-01: MSCA Postdoctoral Fellowships 2022 ExpectedOutcome:

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181.15K€ | Lider

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Duración del proyecto: 35 meses Fecha Inicio: 2023-03-21
Fecha Fin: 2026-02-28

Líder del proyecto

UNIVERSIDAD DE BARCELONA No se ha especificado una descripción o un objeto social para esta compañía.

Total investigadores 327

Presupuesto del proyecto 181K€

Descripción del proyecto Outstanding problems in a variety of mathematical fields have been shown to reduce, in essence, to questions about the combinatorial structure of the transfinite; this was the case, for example, in Shelah's solution to the Whitehead Problem in group theory, Farah's solution to the Brown-Douglas-Fillmore Problem in operator algebras, Solovay and Woodin's solution to Kaplansky's Conjecture on the automatic continuity of Banach algebras, and Moore's solution to the $L$ Space Problem in general topology. At the same time, cardinality and transfinite closing-off arguments play a critical role in many of the most fundamental constructions of contemporary category theory --- in the small object argument, the themes of local presentability and accessibility, or the invocation of Grothendieck universes, for example. Considerations of this latter sort play an insistent role in the 50-year-old problem of the existence of cohomological localizations in the categories of simplicial sets or spectra, a question which the work of this application's sponsor (among others at the University of Barcelona) suggests may well turn out, much as above, to be set theoretic in essence. Work on this problem forms the core of the present proposal, both for its instrinsic interest and bearing on multiple related conjectures (Hovey's Conjecture on Bousfield classes, most prominently), and for its relation, via the phenomenon of $\kappa$-phantom maps, to the researcher's accumulating results on the higher derived limits of large inverse systems. In its course, a number of related questions, for example on the cohomology of small cardinals, will receive close attention as well. This is, in short, a proposal to bring researchers from the divergent fields of algebraic topology and set theory together for work on a well-known problem which they may be uniquely well-suited to solve, and in the process to considerably extend the lines of research for which they are already known.

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