Mathematician Bridges Gap Between Quantum Physics and Number Theory

Mathematicians have long sought to bridge the gap between quantum physics and number theory, two fields that seem worlds apart. Now, researchers led by Thomas Hausel have made a breakthrough using “big algebras” – a novel approach to studying representations of continuous symmetry groups.

Big algebras provide a commutative “mathematical translation” of non-commutative matrix algebra, allowing hidden properties to be revealed geometrically. This work has far-reaching implications for both quantum physics and number theory. Hausel’s research shows that big algebras relate different symmetry groups precisely when their Langlands duals are related, a concept central to the Langlands Program.

The team used algebraic geometry techniques to create 3D-printable shapes that recapitulate sophisticated aspects of the original mathematical information. Daniel Bedats designed and printed these shapes using the Stratasys J750 3D printer at ISTA’s Miba Machine Shop. This work, funded by the Austrian Science Fund, brings us closer to connecting the “continents” of quantum physics and number theory, with potential applications in both fields.

Imagine being able to translate complex mathematical information from one realm to another, unlocking hidden properties and relationships. This is precisely what Dr. Hausel’s work on “big algebras” achieves. By computing the big algebra of a continuous symmetry group, Hausel can represent its essential properties geometrically, creating 3D-printable shapes that recapitulate sophisticated aspects of the original mathematical information.

The significance of this discovery lies in its ability to strengthen the link between quantum physics and number theory. In quantum physics, matrices are used extensively, but they’re often non-commutative, meaning the order of operations affects the result. Big algebras provide a commutative “mathematical translation” of these non-commutative matrix algebras, allowing us to decode and represent their hidden properties geometrically.

Furthermore, Hausel’s work shows that big algebras can reveal relationships between related symmetry groups and their Langlands duals, a central concept in number theory. This has potential applications in bridging the gap between these two mathematical “continents.”

The implications are profound: ideally, big algebras could relate the Langlands duality in number theory with quantum physics, creating a bridge between these two fields. While we’re not there yet, Hausel’s work demonstrates that big algebras can solve problems on both continents, offering a glimpse of the connection between them.

This breakthrough has the potential to revolutionize our understanding of mathematics and its applications. As Dr. Hausel aptly puts it, “With big algebras, the mathematical ‘translation’ does not only work in one direction but in both.” The fog is lifting, and we’re getting a glimpse of the mountains and shores on the horizon, connecting two vast worlds of mathematics.