Periodic Reporting for period 1 - BORCA (Borel combinatorics and Approximations)
Okres sprawozdawczy: 2023-08-01 do 2024-09-30
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 outgoing phase of the fellowship include the following:
(1) Measure reducibility provides a critical framework for the study of complexities of countable Borel equvialence relations (CBERs). One of the central problems in descriptive set theory is to understand the structure of measure reducibility above the Borel hyperfinite CBER E_0. 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 Z^d have seen significant progress in the recent years, for instance, Greenfeld and Tao have disproved the periodic tiling conjecture in high dimensions. It has been demonstrated that translational monotilings in the plane R^2 by axes parallel polygonal sets enjoy similar structure theory as their discrete counterpart in Z^2. This connection has been further used to demonstrate that the corresponding translational monotiling problem in R^2 is decidable.
(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. Employing LOCAL algorithms and network decomposition in the study of Borel hyperfiniteness, initiated by Bernshteyn and Yu in the case of polynomial growth, has proven to be both effective and particularly well-suited for Borel graphs with slow but superpolynomial growth.
(1) Measure reducibility provides a critical framework for the study of complexities of countable Borel equvialence relations (CBERs). One of the central problems in descriptive set theory is to understand the structure of measure reducibility above the Borel hyperfinite CBER E_0. 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 Z^d have seen significant progress in the recent years, for instance, Greenfeld and Tao have disproved the periodic tiling conjecture in high dimensions. It has been demonstrated that translational monotilings in the plane R^2 by axes parallel polygonal sets enjoy similar structure theory as their discrete counterpart in Z^2. This connection has been further used to demonstrate that the corresponding translational monotiling problem in R^2 is decidable.
(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. Employing LOCAL algorithms and network decomposition in the study of Borel hyperfiniteness, initiated by Bernshteyn and Yu in the case of polynomial growth, has proven to be both effective and particularly well-suited for Borel graphs with slow but superpolynomial growth.
The following results of the PI and co-authors have been supported by the project, and the corresponding preprints can be found on arXiv:
(a) With Pikhurko (Warwick), the PI investigates large deviation principles (LDPs) for various random graph models defined through sampling from dense graph limits.
(b) With Higgins (UCLA), the PI has shown that the notion of finite Borel asymptotic dimension, recently utilized in the study of Borel graph combinatorics, is as complex as possible.
(a) With Pikhurko (Warwick), the PI investigates large deviation principles (LDPs) for various random graph models defined through sampling from dense graph limits.
(b) With Higgins (UCLA), the PI has shown that the notion of finite Borel asymptotic dimension, recently utilized in the study of Borel graph combinatorics, is as complex as possible.