Various branches of mathematics have evolved completely independently, each developing its own complex languages that are difficult to interpret. In a recent research published in PNAS, Tamás Hausel, a mathematics professor at the Institute of Science and Technology Austria (ISTA), introduces “big algebras,” a two-way mathematical ‘dictionary’ linking symmetry, algebra, and geometry. This could enhance connections between the seemingly separate realms of quantum physics and number theory.
Technical Toolkit: Symmetry and Commutativity, from Aesthetics to Utility
- Symmetry extends beyond aesthetic appeal and balance; it’s a recurrent concept across various fields. In mathematical terms, symmetry refers to a type of ‘invariance’: a symmetrical object remains unchanged even under certain operations or transformations.
- The collection of all transformations that keep a mathematical object invariant is termed a ‘symmetry group.
- Symmetries such as the rotation of circles or spheres are classified as ‘continuous‘, while ‘discrete’ symmetries include actions like the mirroring of symmetrical objects, such as a butterfly’s wings.
- Continuous symmetry groups are mathematically expressed through matrices—rectangular arrangements of numbers—which can translate the object’s properties into linear algebra.
- Continuous symmetry groups are termed ‘commutative‘ when the sequence of operations doesn’t affect the outcome, and ‘non-commutative‘ when it does.
- For instance, rotations of a circle form a commutative continuous symmetry group. Conversely, the symmetry group of Earth is non-commutative: rotating from Africa left then down yields a different view compared to rotating down then left. In the first sequence, one sees the South Pole, while the latter leads to the equator in the western hemisphere, with the poles positioned horizontally.
- Non-commutative symmetry groups have previously been characterized by non-commutative matrices. However, these do not allow for a geometric interpretation since the spatial aspects of non-commutative algebras remain largely enigmatic. Conversely, commutative algebras are better understood through their geometry.
- A “big algebra” represents a commutative ‘translation’ of a non-commutative matrix algebra, permitting the application of algebraic geometry techniques. Thus, big algebras provide new insights into the characteristics of non-commutative continuous symmetry groups.
Mathematics, recognized for its precision among scientific fields, could be regarded as the quest for absolute truth. Yet, the pathways to mathematical truth often involve overcoming significant challenges, similar to scaling towering mountain peaks or constructing immense bridges across isolated lands. The mathematical landscape is filled with enigmas, as various disciplines have branched out along intricate and separated routes, making it critical to rely on intuition and substantial abstraction. Even foundational elements of physics compel mathematics into greater complexity. This is particularly relevant regarding symmetries, which physicists have utilized to hypothesize and identify numerous subatomic particles constituting our universe.
In a bold initiative, Tamás Hausel, a professor at ISTA, has not only proposed but also validated a novel mathematical entity called “big algebras.” This groundbreaking theorem serves as a ‘dictionary’ interpreting the abstract elements of mathematical symmetry through algebraic geometry. By operating at the confluence of symmetry, abstract algebra, and geometry, big algebras leverage more concrete geometric information to encapsulate sophisticated mathematical concepts related to symmetries. “The big algebras allow us to uncover critical insights into the hidden layers of the intriguing world of symmetry groups from the ‘tip of the mathematical iceberg’,” says Hausel. His work aims to forge a connection between two seemingly distant mathematical realms: “Envision a landscape of quantum physics on one side and, very far off, the realm of number theory. With this research, I hope to bridge these two domains more solidly.”
No Longer Lost in Translation
The philosopher and mathematician René Descartes from the 17th century demonstrated that we could comprehend the geometry of objects through algebraic equations. He was the pioneering figure to ‘translate’ mathematical data between these once-isolated fields. “I tend to conceptualize the relationships among various mathematical fields as dictionaries that translate data between often unintelligible mathematical languages,” comments Hausel. Numerous mathematical ‘dictionaries’ have emerged, but some only convert information in a single direction, keeping the reverse information entirely obscured. Moreover, the current understanding of “algebra” includes both classical algebra, as described in Descartes’ era, and abstract algebra, which examines mathematical structures that are not limited to numerical expressions, adding to the complexity. Hausel now employs abstract algebra alongside algebraic geometry to create a two-way ‘dictionary.’
A Skeleton and Nerves
In mathematics, symmetry is characterized as a form of ‘invariance.’ The collection of transformations that preserves a mathematical object’s state is called a “symmetry group,” which can be categorized as either ‘continuous’ (like rotating a circle or sphere) or ‘discrete’ (such as mirroring an object). Continuous symmetry groups are illustrated through matrices—rectangular arrays of numbers. Utilizing a matrix representation of a continuous symmetry group, Hausel can compute the big algebra and express its key properties geometrically by mapping its ‘skeleton’ and ‘nerves’ onto a mathematical surface. The skeleton and nerves of the big algebra yield intriguing, 3D-printable forms that summarize intricate features of the original mathematical information, completing the translation process. “I am particularly enthusiastic about this work, as it introduces a distinct approach to researching representations of continuous symmetry groups. With big algebras, the mathematical ‘translation’ functions in both directions.”
Connecting Isolated Continents in the Expansive Realm of Mathematics
How can big algebras reinforce the relationship between quantum physics and number theory, two disciplines within mathematics that appear vastly different? Firstly, the mathematics underpinning quantum physics heavily relies on matrices—rectangular arrangements of numbers. However, these matrices are often ‘non-commutative,’ meaning that the multiplication order affects the resulting product. This presents challenges in algebra and algebraic geometry since non-commutative algebra remains partially unexplored. Big algebras address this issue: once computed, a big algebra serves as a commutative ‘mathematical translation’ of a non-commutative matrix algebra. Thus, the information once confined within non-commutative matrices can be deciphered and geometrically illustrated, revealing hidden attributes.
Furthermore, Hausel indicates that big algebras not only clarify connections between related symmetry groups but also in instances where their corresponding “Langlands duals” are related. Langlands duality represents a core concept in the field of number theory. The Langlands Program seeks to establish a comprehensive and intricate dictionary to unify isolated mathematical ‘continents.’ “In my research, it appears that big algebras correlate different symmetry groups precisely when their Langlands duals coincide, a somewhat unexpected finding with promising applications in number theory,” states Hausel.
“In an ideal scenario, big algebras would enable me to establish connections between Langlands duality in number theory and quantum physics,” explains Hausel. For the moment, he has demonstrated that big algebras can resolve issues in both of these disciplines. The fog has begun to lift, allowing the mathematical continents of quantum physics and number theory to glimpse each other’s peaks and shores across the horizon. Soon, rather than merely ferrying between them, a bridge constructed from big algebras could facilitate an easier passage across the mathematical divide that separates these realms.
