HEИндивидуална стипендия2024–2026

CatT · Cohomology and the transfinite

„Хоризонт Европа“ — Действия „Мария Склодовска-Кюри“

Период: 2024-03-01 → 2026-02-28
Финансиране от ЕС: 181 153 €
Участници: 1
Схема: HORIZON-TMA-MSCA-PF-EF

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Накратко на български

Връзката между теорията на множествата и кохомологията се изследва чрез теми като алгебрата на Борел и кохомологията на ординалите. Целта е да се създаде обща основа между различни математически подходи, което помага за по-доброто разбиране на сложни структури.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Cohomology and the transfinite

This project's most overarching ambition is to bring two iconic but largely orthogonal approaches to mathematical objects — the set theoretic and the cohomological — into sustained and mathematically productive dialogue. Its four research units intertwine the study of cohomological structures and of infinitary combinatorics in highly original and mutually illuminating ways: in one representative direction, cohomological perspectives dramatically extend our conception of the higher-dimensional and ZFC combinatorics of small cardinals; in another direction, applications of these same combinatorics in derived and homotopical settings substantially extend our conceptions of their significance. We could also put it like this: relations between the small and large, the local and global, are thematic in both set theory and, say, homological algebra, albeit in, at least superficially, very different ways. A number of recent advances suggest deeper connections between fundamental questions in these two settings, and these form our project's defining concern. Concretely, the project organizes into four interrelated research streams: (1) Cohomological localizations of categories of spaces; (2) The set theory of higher derived limits; (3) Higher walks and the cohomology of the ordinals; (4) Borel definable homological algebra. These streams' constituent problems function, in aggregate, as a laboratory for more methodical minglings of the approaches evoked above, i.e., for developing what have so far been persistent but ad hoc interactions between set theory, homological algebra, and category theory into a robust and programmatically interdisciplinary research field which is of decisive value to each of them. Some of these problems -- as in (1) -- are notorious fifty-year-old open ones; others -- as in portions of (2), (3), and (4) -- reflect such fresh contact between fields as to only really have been conceivable within the past few years.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

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.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITAT DE BARCELONA · BarcelonaКоординаторИспания