Topological aspects of Generalized Descriptive Set Theory

Descripción del proyecto

Conceptos matemáticos avanzados para la informática

El proyecto TopAspOfGDST, que cuenta con el apoyo de las acciones Marie Skłodowska-Curie, aúna dos áreas de las matemáticas: la topología teórica de conjuntos, que estudia formas y espacios, y la teoría descriptiva generalizada de conjuntos (GDST, por sus siglas en inglés), que estudia estructuras matemáticas complejas. La GDST tiene aplicaciones prácticas en campos como la lógica matemática, la informática y los algoritmos. El objetivo del proyecto es fortalecer los vínculos entre la GDST y la topología para poder aplicar herramientas matemáticas consolidadas en la GDST. Ello podría abrir la puerta a importantes avances en ambas áreas y aportar nuevas herramientas teóricas para resolver problemas complejos en lógica, computación y análisis.

Objetivo

"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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