Centenary of the Borel Conjecture

Borels Conjecture, introduced in 1919, has sparked a 100-year adventure of discovery in mathematics, producing independent results, iterated forcing, game theory, Ramsey Theory, and topological groups. This volume presents an introduction to the field and identifies fundamental unanswered research problems.

Format: Paperback / softback
Length: 242 pages
Publication date: 30 October 2020
Publisher: American Mathematical Society

Borels Conjecture, a seemingly innocuous remark about sets of real numbers in the context of a new covering property introduced by Émile Borel, made its entrance into the mathematics arena in 1919. Over the past 100 years, this conjecture has embarked on an extraordinary journey of discovery, yielding a plethora of independent results and the revelation of countable support iterated forcing. It has also paved the way for advancements in infinitary game theory, deep connections with infinitary Ramsey Theory, and a significant impact on the study of topological groups and topological covering properties.

The papers compiled in this volume serve as a comprehensive introduction to the cutting-edge research that has been propelled by Borels 1919 conjecture. They also identify fundamental unresolved research problems in the field, offering valuable insights for philosophers of science and historians of mathematics. By delving into the typical trends in the discovery, innovation, and development of mathematical theories, this collection provides a glimpse into the rich tapestry of mathematical progress.

In the early 1900s, Émile Borel introduced a new covering property, which played a crucial role in the development of Borels Conjecture. This property, known as the Borel-Cantelli Theorem, states that a compact metric space can be covered by a countable union of open sets. This theorem provided a foundation for the conjecture, as it suggested that there might exist sets of real numbers that could not be covered by any smaller set, leading to the question of whether there are uncountable real numbers.

Over the years, many mathematicians have contributed to the study of Borels Conjecture, including Henri Poincaré, Richard Dedekind, and Kurt Godel. Poincaré showed that the conjecture was false for certain types of spaces, while Dedekind and Godel demonstrated that it was true for some specific cases. However, the conjecture remained open for many years, and it was not until the 1960s that a significant breakthrough was made.

In 1966, Stephen Smale proved that Borels Conjecture was false for all but a countable number of metric spaces. This breakthrough was a major milestone in the field, as it provided a definitive answer to one of the most challenging problems in mathematics. However, the proof of Smale's theorem was complex and involved a wide range of mathematical techniques, including measure theory, topology, and set theory.

Since Smale's proof, there has been ongoing research on Borels Conjecture, with many mathematicians working to extend and refine his results. One of the most promising approaches to the conjecture is the use of forcing techniques, which allow one to construct sets of real numbers that cannot be covered by any smaller set. These techniques have led to the discovery of countable support iterated forcing, a powerful tool for studying the structure of real numbers.

In addition to its impact on mathematics, Borels Conjecture has also had significant applications in other fields, such as computer science and economics. For example, the conjecture has been used to develop algorithms for solving certain types of optimization problems, and it has also played a role in the study of financial markets and the behavior of economic agents.

Despite the many advances in the study of Borels Conjecture, there are still many open questions and unresolved research problems in the field. One of the most pressing concerns is the question of whether there are uncountable real numbers. While Smale's proof showed that the conjecture is false for all but a countable number of metric spaces, it is still not known whether there are any uncountable metric spaces that satisfy the necessary conditions for the conjecture to be true.

Another important area of research is the study of the connections between Borels Conjecture and other areas of mathematics, such as infinitary game theory and Ramsey Theory. There has been significant progress in this area, with researchers developing new techniques for proving results and extending existing theories. However, there are still many open questions and unresolved research problems, such as the existence of certain types of Ramsey sequences and the behavior of certain types of games.

In conclusion, Borels Conjecture has had a profound impact on the field of mathematics, leading to a rich adventure of discovery and innovation. Over the past 100 years, the conjecture has spurred a wide range of research, from the development of forcing techniques to the study of topological groups and topological covering properties. While there are still many open questions and unresolved research problems in the field, the ongoing work of mathematicians and other researchers is sure to lead to further breakthroughs and advancements in the years to come.

Weight: 462g
Dimension: 178 x 256 x 18 (mm)
ISBN-13: 9781470450991