Borel combinatorics and Approximations - Jan Grebík (BORCA)
Infinite graphs and their combinatorics model large real-life networks, like the internet or particles in a crystal, but are also an essential tool to understand mathematical structures that are intrinsically infinite, like the geometry of the Euclidean spaces. The project concerns research on the boundary between logic, analysis and combinatorics, more specifically, research in descriptive set theory and its interactions with measure theory, dynamical systems, computer science and graph limits, through the study of regularity properties of combinatorial problems on infinite graphs. These questions are intimately connected with the theory of distributed computing, random processes and group theory. The recently discovered formal connections between these fields, the new ideas from geometric group theory, or the new insights on the determinacy method have already found many applications and promise to gain new perspectives on old problems. We propose to employ, combine and further develop these methods with particular emphasis on applications to the study of central questions of descriptive set theory, that is, Borel hyperfiniteness, equidecomposition problems, or the abstract classification problem, as well as on finding new links and applications to classical graph theory, in particular, to algorithmic aspects of partition problems on finite graphs.
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