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Quanta Magazine; source: Giles Gardam

### Mathematician Disproves 80-Year-Old Algebra Conjecture

Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

www.quantamagazine.org

On February 22, a postdoctoral mathematician named Giles Gardam gave an hourlong online talk about the unit conjecture, a basic but confounding algebra question that had stood open for more than 80 years. He carefully laid out the history of the conjecture and two allied conjectures, and explained their connections to the powerful algebraic machinery called K-theory. Then, in the final minutes of his presentation, he delivered the kicker.

“I’m nearly at the end of the talk, and it’s time for me to tell you what’s new,” he said. “I’m really happy to be able to announce today, for the first time, that in fact the unit conjecture is false.”

Gardam declined to tell the audience just how he had found the long-sought-after counterexample (except to confirm that it involved a computer search). He would share more details in a few months, he told Quanta. But for now, he said, “I’m still optimistic that maybe I have enough tricks left to get some more results.”

The problem Gardam solved concerns a question simple enough to explain to high school students: Within a broad family of algebraic structures, which elements have multiplicative inverses?

Multiplicative inverses are pairs, like 7 and 1/7, that multiply out to 1. But the unit conjecture concerns multiplicative inverses not of ordinary numbers but of elements in a “group algebra,” a structure that combines a number system (like the real numbers, or certain clock-style arithmetics) with a group (a broad category that includes collections of matrices, symmetry transformations and many other objects).

Within such a structure, mathematicians conjectured more than eight decades ago, only the simplest elements can have multiplicative inverses. Researchers in the middle of the 20th century used extensive paper-and-pencil computations to comb through these group algebras searching for more complicated elements with multiplicative inverses, but they could neither prove the conjecture nor turn up a counterexample.

Over the decades, the unit conjecture and two allied conjectures came to be “seen as hopeless things,” said Dawid Kielak, of the University of Oxford. But even after many mathematicians gave up on proving the three conjectures, they remained “always somehow in the background” of algebraic research, he said, thanks in large part to their deep connections with K-theory.

Now Gardam, of the University of Münster, has disproved the unit conjecture by finding unusual “units” — elements with multiplicative inverses — inside a group algebra built out of the symmetries of a particular three-dimensional crystallographic shape. “It’s a fabulous piece of work,” said Peter Kropholler, of the University of Southampton.

Prior to Gardam’s work, in the absence of either a counterexample or an all-encompassing proof, mathematicians had plugged away at establishing the three conjectures (or some of their downstream consequences) in special cases. Often this involved tapping the powerful but laborious machinery of K-theory. Gardam’s discovery of a counterexample to the unit conjecture is oddly reassuring, Kropholler said, because it suggests that this hard work was really needed.

“At the very basic core there was always this nagging question: If you just had a proof of the unit conjecture, wouldn’t that make a lot of things a lot easier?” he said. Knowing that the conjecture is not universally true, he said, means that “all the complicated things we did to avoid having to find a proof of the unit conjecture were still very worth doing.” Researchers are now tasked with understanding the principles behind Gardam’s complicated units. “It’s very exciting,” Kielak said. “We’re at this moment where the floodgates opened and now everything is possible again.”

*Unpredictable Cancellations*

The unit conjecture draws upon the vast universe of group theory, which studies sets that have some notion of how to “multiply” two elements to get a new one. As long as the multiplication operation is reasonably well behaved, there are just two additional requirements for a set to qualify as a group: The set must contain a special element (usually labeled “1”) which leaves other elements unchanged when multiplied with them, and every element g must have a multiplicative inverse (written g−1), such that g times g−1 equals 1. (It’s not until we move into the realm of a group algebra, which combines the group with a coefficient number system, that elements crop up that lack multiplicative inverses, and the unit conjecture comes into play.)

The world of groups is immense: There are groups of matrices (arrays of numbers) and groups of symmetry transformations, groups that keep track of the number of holes inside a shape or the different arrangements of a deck of cards, and groups that arise in physics and cryptography and a host of other domains.

In many groups, there’s only one arithmetic operation that makes sense. But matrices are different: Besides multiplying them, you could also add them, or multiply a matrix by a numerical coefficient. Matrices are the key to understanding linear objects and transformations, and because of this power, mathematicians and physicists often gain insight into other groups by finding ways to represent the group elements as matrices.

About a century ago, group theorists started asking: If we’re going to represent the elements of a group as matrices, why not encapsulate some of the special properties of matrices within the structure of the original group? In particular, why not talk about adding together group elements or multiplying them by coefficients from some number system? After all, if a and b are two group elements, it’s possible at least to write down sums like ½*a + 7*b or 4a3 − 2ab2.

These sums often have no meaning in terms of the original group — it doesn’t make sense to talk about one-half of an arrangement of a deck of cards plus seven times another arrangement. But you can nevertheless carry out algebraic manipulations on these formal sums. Mathematicians call the collection of these formal sums a “group algebra,” and this structure, which weaves together the group and a coefficient number system, “packs together information about the [matrix] representations of [the group] in one object,” Gardam wrote in an email.