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Topological aspects of Generalized Descriptive Set Theory

Project description

Advancing mathematical concepts for computer science

With the support of the Marie Skłodowska-Curie Actions programme, the TopAspOfGDST project brings together two areas of mathematics: set-theoretic topology, which studies shapes and spaces, and generalised descriptive set theory (GDST), which looks at complex mathematical structures. GDST can have practical applications in areas such as mathematical logic and computer science and algorithms. The aim of the project is to create stronger connections between GDST and topology, allowing established mathematical tools to be applied in GDST. This could lead to new breakthroughs in both fields and provide new theoretical tools for solving complex problems in logic, computation, and analysis.

Objective

"My project lies at the intersection of set-theoretic topology and generalized descriptive set theory (GDST) on uncountable cardinals. Since the 1950s, topology has extensively studied metric spaces and their generalizations, but as foundational questions were largely resolved, research shifted toward new areas. In contrast, GDST is a relatively recent field that has gained considerable interest over the past two decades. Both fields share a common interest in studying (subsets of) the generalized Baire space, but they typically use distinct tools and tackle different questions.

Recently, efforts have been made to bridge these two areas, by introducing appropriate classes of ""Polish-like"" spaces in GDST, allowing the application of established topological tools in GDST while also opening new perspectives in topology. This synergy promises exciting opportunities for innovative research and greater insight in both fields and generates new applications of GDST beyond the well-established ones. However, these efforts have been scattered, with different classes of spaces being used, and they are still in their early stages, leaving much to be explored.

My project aims to strengthen this connection by systematically studying and comparing the various approaches used to incorporate topology into GDST, to establish a unified topological foundation for GDST. In the process, I will also tackle several open questions in topology that have arisen from recent GDST applications.

The project has two primary research goals. The first focuses on extending notions equivalent to metrizability to uncountable cardinals, including concepts like Nagata-Smirnov or Bing bases, uniform spaces, and regular bases derived from Arhangel'skii's Metrization Theorem. The second goal addresses completeness notions for topological spaces without a metric, such as (long) Choquet games, Čech-completeness, and the completeness of uniformities.
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Topic(s)

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Funding Scheme

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HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships

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Call for proposal

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(opens in new window) HORIZON-MSCA-2024-PF-01

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Coordinator

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Net EU contribution

Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.

€ 166 903,92
Address
REALTANODA STREET 13-15
1053 Budapest
Hungary

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Region
Közép-Magyarország Budapest Budapest
Activity type
Other
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Total cost

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