Several fields of mathematics have developed in total isolation, using their very own ‘undecipherable’ coded languages. In a brand new study published in PNAS, Tamás Hausel, professor of mathematics on the Institute of Science and Technology Austria (ISTA), presents “big algebras,” a two-way mathematical ‘dictionary’ between symmetry, algebra, and geometry, that would strengthen the connection between the distant worlds of quantum physics and number theory.
Technical Toolkit: Symmetry and commutativity, from esthetics to functionality
- Symmetry isn’t just a matter of esthetics and equilibrium, but additionally a highly recurrent feature throughout the domains of life. Mathematically, symmetry is a type of ‘invariance‘: Even when subjected to certain operations or transformations, a symmetrical mathematical object stays unchanged.
- The group of all transformations under which a mathematical object stays invariant is named a ‘symmetry group.’
- Symmetries similar to the rotation of a circle or a sphere will be classified as ‘continuous.’ (Contrary to this, an example of a ‘discrete‘ symmetry is the mirroring of a bilaterally symmetrical object like a butterfly’s wings.)
- Continuous symmetry groups are mathematically represented by matrices — rectangular arrays of numbers — which may convert the properties of the mathematical object into linear algebra.
- Continuous symmetry groups are called ‘commutative‘ when the order of the operations or transformations doesn’t matter, or ‘non-commutative‘ in the alternative case.
- Rotations of a circle will be seen as a commutative continuous symmetry group. In contrast, the symmetry group of the planet Earth is non-commutative: if one starts by taking a look at the equator going through Africa, rotating left then down doesn’t produce the identical result as rotating down then left. In the primary case, one’s view could be centered on the South Pole. Within the second, one would arrive on the equator within the western hemisphere with the poles positioned horizontally.
- Non-commutative symmetry groups have been to this point represented by non-commutative matrices, i.e., matrices during which the order of operations influences the tip result. Nevertheless, this doesn’t allow a geometrical interpretation because the geometry of non-commutative algebras isn’t yet well understood. However, commutative algebras will be well understood through their geometry.
- A “big algebra” is a commutative ‘translation’ of a non-commutative matrix algebra, and thus allows using algebraic geometry techniques. Because of this, big algebras shed latest light on the properties of non-commutative continuous symmetry groups.
Mathematics, essentially the most exact amongst scientific disciplines, could possibly be viewed as the final word quest for absolute truth. Nevertheless, the mathematical roads to truth often must overcome tremendous obstacles, very similar to conquering unimaginably high mountain peaks or constructing giant bridges between isolated continents. The mathematical world abounds with mysteries and a number of other mathematical disciplines have developed along convoluted paths — in complete isolation from each other. Thus, establishing an irrefutable truth around complex phenomena within the physical world draws on intuition and deal of abstraction. Even fundamental elements of physics push mathematics to latest heights of complexity. This is very true for symmetries, with the assistance of which physicists have theorized and discovered a whole zoo of subatomic particles that make up our universe.
In an exceptionally ambitious endeavor, Tamás Hausel, professor on the Institute of Science and Technology Austria (ISTA), not only conjectured but additionally proved a brand new mathematical tool called “big algebras.” This latest theorem is comparable to a ‘dictionary’ that deciphers essentially the most abstract elements of mathematical symmetry using algebraic geometry. By operating on the intersection of symmetry, abstract algebra, and geometry, big algebras use more tangible geometric information to recapitulate sophisticated mathematical details about symmetries. “With big algebras, information from the ‘tip of the mathematical iceberg’ may give us unprecedented insights into the hidden depths of the mysterious world of symmetry groups,” says Hausel. With this mathematical breakthrough, Hausel seeks to consolidate the connection between two distant fields of mathematics: “Imagine, on the one hand, a world of mathematical representations of quantum physics, and alternatively, very, very distant, the purely mathematical world of number theory. With the current work, I hope to have come one step closer to establishing a stable connection between these two worlds.”
Not lost in translation
The 17th-century philosopher and mathematician René Descartes showed us that we could understand the geometry of objects by utilizing algebraic equations. Thus, he was the primary to ‘translate’ mathematical information between these previously separate fields. “I wish to view the relations between different mathematical fields as dictionaries that translate information between often non-mutually intelligible mathematical languages,” says Hausel. To this point, several such mathematical ‘dictionaries’ have been developed, but some only translate the data in a single direction, leaving the data concerning the way back entirely encrypted. Moreover, the term “algebra” nowadays encompasses each classical algebra, as in Descartes’ time, and abstract algebra, i.e. the study of mathematical structures that can’t necessarily be expressed with numerical values. This adds one other layer of complexity. Now, Hausel uses abstract algebra and algebraic geometry as a two-way ‘dictionary’.
A skeleton and nerves
In mathematics, symmetry is defined as a type of ‘invariance’. The group of transformations that keep a mathematical object unchanged is named a “symmetry group.” These are classified as ‘continuous’ (e.g., the rotation of a circle or sphere) or ‘discrete’ (e.g., the mirroring of an object). Continuous symmetry groups are represented mathematically by matrices — rectangular arrays of numbers. Ranging from a matrix representation of a continuous symmetry group, Hausel can compute the massive algebra and represent its essential properties geometrically by drawing its ‘skeleton’ and ‘nerves’ on a mathematical surface. The large algebra’s skeleton and nerves give rise to interesting, 3D-printable shapes that recapitulate sophisticated elements of the unique mathematical information, thus closing the interpretation circle. “I’m particularly enthusiastic about this work, because it provides us with a very novel approach to studying representations of continuous symmetry groups. With big algebras, the mathematical ‘translation’ doesn’t only work in a single direction but in each.”
Bridging isolated continents in an unlimited world of mathematics
How could big algebras strengthen the link between quantum physics and number theory, two fields of mathematics seemingly worlds apart? Firstly, the maths behind quantum physics makes extensive use of matrices — rectangular arrays of numbers. Nevertheless, these matrices are typically ‘non-commutative,’ meaning that multiplying the primary matrix by the second doesn’t yield the identical result as multiplying the second by the primary. This poses an issue in algebra and algebraic geometry as non-commutative algebra isn’t yet well understood. Big algebras now solve this problem: when computed, an enormous algebra is a commutative ‘mathematical translation’ of a non-commutative matrix algebra. Thus, the data initially enclosed inside non-commutative matrices will be decoded and represented geometrically to disclose their hidden properties.
Secondly, Hausel shows that big algebras not only reveal relationships between related symmetry groups, but additionally when their so-called “Langlands duals” are related. These duals are a central concept within the purely mathematical world of number theory. Within the Langlands Program, a highly intricate, large-scale dictionary that seeks to bridge isolated mathematical ‘continents,’ the Langlands duality is an idea or tool that permits ‘mapping’ mathematical information between different categories. “In my work, big algebras appear to relate different symmetry groups precisely when their Langlands duals are related, a quite surprising end result with possible applications in number theory,” says Hausel.
“Ideally, big algebras would allow me to relate the Langlands duality in number theory with quantum physics,” says Hausel. For now, he was capable of show that big algebras solve problems on each of those continents. The fog has began to dissipate, and the continents of quantum physics and number theory have caught a glimpse of every others’ mountains and shores on the horizon. Soon, slightly than only connecting the continents by boat, a bridge of massive algebras might allow a neater crossing of the mathematical strait separating them.
