Periodic Reporting for period 2 - BORCA (Borel combinatorics and Approximations)
Periodo di rendicontazione: 2024-10-01 al 2025-09-30
This project concerns research on the boundary between analysis, logic and combinatorics, more specifically, investigates problems in descriptive set theory and their interactions with measure theory, dynamical systems, computer science and graph limits, through the study of measurability properties of combinatorial problems on infinite graphs.
The basic objects of study in descriptive set theory are constructible (definable) subsets of the real line, more abstractly, objects that can be encoded with a countably infinite amount of information. Such objects are ample throughout mathematics which makes descriptive set theory a versatile tool for combining various techniques and perspectives from different areas, and providing connections between seemingly unrelated problems. An important aspect of combinatorial problems on finite graphs, motivated by applications in computer science, is to understand when there is an efficient algorithm that computes a solution. Embracing these algorithmic aspects of graph problems in the study of infinite graphs in recent years led to novel methods and breakthrough results. The main goal of this project will be to exploit and further develop this connection with particular emphasis on applications to the study of the central questions of descriptive set theory.
Particular topics that are studied in the project concern problems around approximations of analytic objects such as (Borel) hyperfiniteness and graph limits, as well as some classical tiling problems in Euclidean spaces.
The basic objects of study in descriptive set theory are constructible (definable) subsets of the real line, more abstractly, objects that can be encoded with a countably infinite amount of information. Such objects are ample throughout mathematics which makes descriptive set theory a versatile tool for combining various techniques and perspectives from different areas, and providing connections between seemingly unrelated problems. An important aspect of combinatorial problems on finite graphs, motivated by applications in computer science, is to understand when there is an efficient algorithm that computes a solution. Embracing these algorithmic aspects of graph problems in the study of infinite graphs in recent years led to novel methods and breakthrough results. The main goal of this project will be to exploit and further develop this connection with particular emphasis on applications to the study of the central questions of descriptive set theory.
Particular topics that are studied in the project concern problems around approximations of analytic objects such as (Borel) hyperfiniteness and graph limits, as well as some classical tiling problems in Euclidean spaces.
The main topics addressed by the PI during the course of the fellowship include the following:
(1) Measure reducibility provides a critical framework for studying the complexities of countable Borel equivalence relations (CBERs). One of the central questions in descriptive set theory is to understand the structure of measure reducibility above the Borel hyperfinite CBER E₀. Newly discovered connections with the theory of random graphs and Lie groups suggest the potential to leverage advanced techniques from these areas in the study of measurable dynamics.
(2) Translational monotilings of Zd have seen significant progress in recent years; for instance, Greenfeld and Tao disproved the periodic tiling conjecture in high dimensions. It has been shown that translational monotilings of the plane R² by axis-parallel polygonal sets exhibit a structure theory analogous to their discrete counterparts in Z². This connection has further been used to devise a simple procedure that decides whether a given axis-parallel polygonal set tiles R².
(3) A general approach to solving local coloring problems on graphs, widely used in mathematics and theoretical computer science, involves decomposing graphs into suitable finite pieces where the problem can be solved efficiently. The use of LOCAL algorithms and network decomposition in the study of Borel hyperfiniteness, initiated by Bernshteyn and Yu in the case of polynomial growth, has proven both effective and particularly well-suited for Borel graphs with slow but superpolynomial growth.
(4) Connections between the theory of cost and percolation have been known for a long time. Recently, Fraczyk, Mellick, and Wilkens showed that cocompact lattices in higher-rank semisimple real Lie groups have fixed price one. Building on these insights, we investigate the classical Poisson–Voronoi percolation model in higher-rank symmetric spaces and product spaces.
(1) Measure reducibility provides a critical framework for studying the complexities of countable Borel equivalence relations (CBERs). One of the central questions in descriptive set theory is to understand the structure of measure reducibility above the Borel hyperfinite CBER E₀. Newly discovered connections with the theory of random graphs and Lie groups suggest the potential to leverage advanced techniques from these areas in the study of measurable dynamics.
(2) Translational monotilings of Zd have seen significant progress in recent years; for instance, Greenfeld and Tao disproved the periodic tiling conjecture in high dimensions. It has been shown that translational monotilings of the plane R² by axis-parallel polygonal sets exhibit a structure theory analogous to their discrete counterparts in Z². This connection has further been used to devise a simple procedure that decides whether a given axis-parallel polygonal set tiles R².
(3) A general approach to solving local coloring problems on graphs, widely used in mathematics and theoretical computer science, involves decomposing graphs into suitable finite pieces where the problem can be solved efficiently. The use of LOCAL algorithms and network decomposition in the study of Borel hyperfiniteness, initiated by Bernshteyn and Yu in the case of polynomial growth, has proven both effective and particularly well-suited for Borel graphs with slow but superpolynomial growth.
(4) Connections between the theory of cost and percolation have been known for a long time. Recently, Fraczyk, Mellick, and Wilkens showed that cocompact lattices in higher-rank semisimple real Lie groups have fixed price one. Building on these insights, we investigate the classical Poisson–Voronoi percolation model in higher-rank symmetric spaces and product spaces.
The following results of the PI and co-authors have been supported by the project, and the corresponding preprints are available on arXiv:
(a) With Pikhurko (Warwick), we investigate large deviation principles (LDPs) for various random graph models defined via sampling from dense graph limits.
(b) With Higgins (UCLA), we show that the notion of finite Borel asymptotic dimension, recently utilized in the study of Borel graph combinatorics, is maximally complex.
(c) With Z. Vidnyánszky (ELTE), we survey recent connections between descriptive set theory and distributed computing, emphasizing how techniques from the former field influence the latter.
(d) With C. Ikenmeyer and O. Pikhurko (Warwick), we show that fractional divisibility of the sphere by r rotations is impossible if at least r/2 of the rotations are generic.
(e) With K. Recke (Münster), we show that the uniqueness thresholds for Poisson–Voronoi percolation in symmetric spaces of connected higher-rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit.
(f) With D. Král' (Leipzig), X. Liu (Warwick), O. Pikhurko (Warwick), and J. Slipantschuk (Bayreuth), we show that the spectra of a convergent sequence of dense digraphs converge to the spectrum of the limiting digraphon.
(a) With Pikhurko (Warwick), we investigate large deviation principles (LDPs) for various random graph models defined via sampling from dense graph limits.
(b) With Higgins (UCLA), we show that the notion of finite Borel asymptotic dimension, recently utilized in the study of Borel graph combinatorics, is maximally complex.
(c) With Z. Vidnyánszky (ELTE), we survey recent connections between descriptive set theory and distributed computing, emphasizing how techniques from the former field influence the latter.
(d) With C. Ikenmeyer and O. Pikhurko (Warwick), we show that fractional divisibility of the sphere by r rotations is impossible if at least r/2 of the rotations are generic.
(e) With K. Recke (Münster), we show that the uniqueness thresholds for Poisson–Voronoi percolation in symmetric spaces of connected higher-rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit.
(f) With D. Král' (Leipzig), X. Liu (Warwick), O. Pikhurko (Warwick), and J. Slipantschuk (Bayreuth), we show that the spectra of a convergent sequence of dense digraphs converge to the spectrum of the limiting digraphon.