A significant breakthrough in mathematics has emerged as Anton Bernshteyn, a researcher at the University of California, Los Angeles, unveiled a surprising connection between the realms of descriptive set theory and computer science. In 2023, Bernshteyn demonstrated that challenges surrounding certain types of infinite sets can be reframed as problems concerning the communication networks of computers. This unexpected link has opened new avenues for collaboration between mathematicians and computer scientists, reshaping the landscape of both disciplines.
Descriptive set theory, a niche area of mathematics, focuses on the intricate nature of sets, particularly those involving infinity, which are often overlooked by mainstream mathematicians. According to Václav Rozhoň, a computer scientist at Charles University in Prague, this development is “something really weird” and challenges the traditional separations between mathematical fields. The two domains typically employ distinct languages—logic for set theorists and algorithms for computer scientists—raising questions about the equivalence of their problems.
Since Bernshteyn’s revelation, researchers have actively sought to traverse this new bridge, aiming to apply insights from computer science to generate new theorems in descriptive set theory. In a notable shift, some set theorists are reconsidering how they understand infinity by incorporating concepts from computer science. “We’ve been working on very similar problems without directly talking to each other,” noted Clinton Conley, a descriptive set theorist from Carnegie Mellon University.
Bernshteyn’s journey into descriptive set theory began during his undergraduate studies, where he encountered the subject as a seemingly obsolete field. However, his perspective changed during a logic course with Anush Tserunyan, who became one of his mentors. Tserunyan emphasized the significance of logic and set theory as a unifying framework within mathematics, sparking Bernshteyn’s interest in the field.
The roots of descriptive set theory trace back to the work of Georg Cantor, who, in 1874, established that there are various sizes of infinity. This concept remains central to the field, which classifies sets based on their complexity and measurability. A set’s “measure” quantifies its size in terms of length, area, or volume, differing from Cantor’s “cardinality,” which counts the elements. Descriptive set theorists organize sets into hierarchies based on their measurability, with simpler sets at the top and “unmeasurable” sets at the bottom.
Bernshteyn focuses on infinite graphs—collections of nodes connected by edges—which are often neglected by graph theorists who typically examine finite graphs. These infinite structures have significant implications for understanding dynamical systems and other mathematical phenomena. For example, Bernshteyn studies how to color the nodes of these graphs under specific constraints, a problem that requires collaboration with computer scientists to find efficient solutions.
One illustrative problem involves creating colorings for infinite graphs where nodes represent Wi-Fi routers in a network, ensuring that no two connected routers operate on the same frequency channel. In 2019, Bernshteyn attended a computer science talk that fundamentally shifted his research trajectory. The discussion revolved around “distributed algorithms,” which operate without a central coordinator and can be framed as coloring problems on graphs.
Realizing that the thresholds for coloring problems in computer science mirrored those in descriptive set theory, Bernshteyn sought to formalize this connection. He proposed that every efficient local algorithm could correspond to a measurable way of coloring an infinite graph, thereby establishing a profound relationship between the two fields. This assertion surprised many mathematicians, illustrating a deep link between computation and definability.
Since Bernshteyn’s findings, mathematicians have been eager to explore the implications of this connection. For instance, Rozhoň and colleagues recently applied insights from computer science to color special graphs known as trees, shedding light on the tools required to study corresponding dynamical systems.
The bridge created by Bernshteyn not only provides a new toolkit for solving mathematical problems but also enhances the understanding of set theory as a dynamic field. By integrating concepts from computer science, set theorists can now better classify previously ambiguous problems. Bernshteyn aims to change the perception of set theory, encouraging mathematicians to view it as relevant and interconnected with practical applications. “I want people to get used to thinking about infinity,” he stated, underscoring his commitment to advancing this essential area of research.
