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.

Introduction

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 $latex \frac{1}{7},$ 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 $latex \frac{1}{2}$a + 7b 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.

In many ways, the elements in a group algebra resemble the familiar polynomials from high school algebra: expressions like x2 − 4x + 5 or 3x3y5 + 2. But there’s a key difference. If you multiply two polynomials, some terms might cancel out, but the term with the highest exponent will always survive the cancellation process. For example, (x − 1)(x + 1) = x2 + xx − 1, and while the x and −x terms cancel each other out, the x2 term survives (as does the −1), to produce x2 − 1. But in a group algebra, the relationships between group elements can lead to additional, hard-to-predict cancellations.

For example, suppose our group is the collection of symmetry transformations of the letter “A.” This group has only two elements: the transformation that leaves every point where it is (the “1” in our group), and the reflection across the central vertical axis (let’s call this reflection r). Reflecting twice restores each point to its original location, so in the language of our group multiplication, r times r equals 1. This relationship leads to all sorts of unexpected outcomes in the group algebra — for example, if you multiply r + 2 with $latex \frac{-r}{3}$ + $latex \frac{2}{3}$, nearly everything cancels out and all that’s left is 1:

latex \begin{aligned}(r + 2)\big(\frac{-r}{3} + \frac{2}{3}\big) &= \frac{-r^2}{3} + \frac{2r}{3} – \frac{2r}{3}+\frac{4}{3}\\[1pt] \\&= \frac{-r^2}{3} + \frac{4}{3}\\[1pt] \\&= 1\hspace{1mm} (\text {since}\hspace{1mm}r^2 = 1)\end{aligned}

In other words, r + 2 and $latex \frac{-r}{3}$ + $latex \frac{2}{3}$ are multiplicative inverses.

In 1940, an algebraist named Graham Higman made a daring conjecture in his doctoral thesis: The worst of this cancellation weirdness, he proposed, will only happen if the group that is used to construct the group algebra contains elements for which some power equals 1, as with r in the example above. In all other group algebras, he posited, while elements with just a single term, like 7a or 8b, can (and do) have multiplicative inverses, sums with multiple terms like r + 2 or 3r − 5s can never have multiplicative inverses. Since elements with multiplicative inverses are called units, Higman’s hypothesis came to be known as the unit conjecture.

Over the next few decades, Irving Kaplansky, one of the leading mathematicians of the 20th century, popularized this conjecture along with two other group algebra conjectures called the zero divisor and idempotent conjectures; the three came to be known as the Kaplansky conjectures. Collectively, the three conjectures posit that group algebras are not too radically different from the algebra we’re used to from multiplying numbers or polynomials. But although Kaplansky called attention to these conjectures, there’s no particular reason to think he believed them, Kielak said.

At the time, there was little evidence either way. If anything, there was a philosophical reason to disbelieve the conjectures: As the mathematician Mikhael Gromov is said to have observed, the menagerie of groups is so diverse that any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.

So for Kaplansky to promote the unit conjecture was “very audacious,” Kielak said. It was “meant to provoke other people to come up with clever examples,” he said.

But mathematicians couldn’t come up with counterexamples, and not for want of trying. In the absence of a counterexample, Kielak said, “you start to think that there’s something deeper going on — that there is some underlying principle that we’ve missed.”

Collapsing Sums

Over the second half of the 20th century, a candidate for that “something deeper” seemed to emerge: algebraic K-theory, a vast edifice that uses difficult-to-compute group invariants to unite algebra with a wide range of mathematical disciplines, such as topology and number theory. Using K-theory, for instance, researchers were able to connect the unit conjecture to the question of when a topological shape can be converted into another shape using only prescribed moves.

Researchers were able to show that certain powerful but unproved K-theory conjectures would imply the zero divisor and idempotent conjectures, potentially offering up a deep reason why they might be true. But they couldn’t do the same for the unit conjecture, the strongest of the three. Wolfgang Lück, of the University of Bonn, tried hard to prove that the unit conjecture follows from a K-theory conjecture called the Farrell-Jones conjecture. “I was never able to make this proof,” he said. “I was wondering whether I’m stupid.”

Mathematicians were nevertheless able to prove the unit conjecture for many specific classes of groups by showing that those groups had a property akin to the notion of the highest exponent in polynomials. But researchers also knew of a handful of groups that violate this property, including a simple one called the Hantzsche-Wendt group. This group captures the symmetries of a shape physicists have considered as a possible model for the shape of the universe, and which is built by gluing together the sides of a three-dimensional crystal. Compared to many other groups, this one is “remarkably unexotic,” said Timothy Riley, of Cornell University.

The Hantzsche-Wendt group seemed like a fruitful place to search for a counterexample to the unit conjecture. But doing so was no straightforward task: The Hantzsche-Wendt group is infinite, so there are infinitely many possibilities even for short sums in the group algebra. And in 2010, a pair of mathematicians showed that if there is a counterexample in this group, it will not be found among the simplest of these sums.

Now Gardam has turned up a pair of multiplicative inverses with 21 terms each within a group algebra built from the Hantzsche-Wendt group. Finding the pair required a complex computer search, but verifying that they really are inverses is well within the realm of human computation. It’s simply a matter of multiplying them together and checking that the 441 terms in the product simplify down to the number 1. “Everything collapses down,” Kropholler said. “That’s pretty amazing.”

Lück knows now why he was never able to prove that the Farrell-Jones conjecture implies the unit conjecture: The Farrell-Jones conjecture is true for the Hantzsche-Wendt group, but the unit conjecture is false. “Now I know I wasn’t stupid,” he said.

Once Gardam releases the details of his algorithm, it will be open season for other mathematicians to explore the Hantzsche-Wendt group and potentially other groups. “The hope is that we will learn something new — a new trick which will allow us to build examples,” Kielak said.

Already, knowing that the conjecture is false has changed the mindsets of many mathematicians. “Psychologically, this is a very big difference,” Kielak said. “Probably in a year’s time, we’re going to have infinitely many” counterexamples.

Gardam’s counterexample uses one of the simplest number systems for its coefficients, a clock arithmetic with only two “hours.” So one immediate question is whether there are counterexamples to be found using other number systems such as the real or complex numbers. There’s also the question of whether some group exists that violates Kaplansky’s other two conjectures. Such a find would send shudders through the K-theory community, since it would contradict some of the subject’s central conjectures.

For Gardam, his discovery is the culmination of years spent hunting for intriguing counterexamples in algebra. He’s not motivated by a bounty hunter’s mentality, he explained in an email — rather, he chases after the frisson of delight that curious counterexamples can give.

“Powerful theory has its own beauty and elegance, but if everything is rigid, tightly controlled and well behaved, the subject can get very dry,” he wrote. “Surprising examples are a big part of what makes maths fun and keeps it weird and wonderful.”