We've come to the end of 2022, making it time to take stock of the biggest math stories of the year that was!

In choosing these stories, we've emphasized math stories involving practical applications in selecting the contenders for the Biggest Math Story of 2022, which we'll present at the end of this year's edition. But before we get started, let's take a minute to do something mathematical that has no direct practical application but which is surprisingly satisfying by watching the following video visualizing the first million integers and coloring them by their prime factors.

There is a practical application in John Williamson's integer mapping exercise, which is the UMAP data visualization tool used to make this short video. UMAP is the acronym for Uniform Manifold Approximation and Projection, which can be applied to large datasets to help make better sense of the information within them in the growing field of visual analytics.

This introduction is already working on several different levels. Intentionally. Most obviously, it's the beginning of the longest article we write each year. It's also provided the opportunity to introduce a practical application that came about from maths that were originally developed as little more than intellectual curiosities. Maths that weren't very practical until the applications they enabled made them essential. The Biggest Math Story of 2022 involves what we'll describe in this introduction as the ultimate application arising from what some describe as most unlikely math.

We'll get there soon enough. Because the Biggest Math Story of 2022 serves as our final article of the year, we've written it to be read and revisited as you might like throughout the holidays. You can take your time to explore the stories and follow the links to go deeper into the various topics. Or you can just blow past all the sections and go straight to the Biggest Math Story of 2022 so you have something extra smart to talk about at holiday parties. Let's get to it....

Artificial Intelligence Discovers Faster Math

Picking up from where we left off with the biggest math story of 2021, artificial intelligence continued to make inroads in powering mathematical discoveries, even as the technology made bigger headlines during the year in other fields, including in art and writing.

Perhaps the most notable accomplishment after last year's major achievements by the DeepMind team was the discovery of a faster, more efficient method for multiplying matrices. And they did it by making a game out of doing the math:

The DeepMind team approached the problem by turning tensor decomposition into a single-player game. They started with a deep learning algorithm descended from AlphaGo — another DeepMind AI that in 2016 learned to play the board game Go well enough to beat the top human players.

Using that base, they developed a new agent called AlphaTensor to play their matrix multiplication game. The DeepMind team summarized their approach in the paper announcing their accomplishment:

Here we report a deep reinforcement learning approach based on AlphaZero for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago.

That has the makings of being a big deal, but as the researchers acknowledge, their accomplishment is limited in its utility:

Researchers also emphasized that immediate applications of the record-breaking 4-by-4 algorithm would be limited: Not only is it valid only in modulo 2 arithmetic, but in real life there are important considerations besides speed.

Here's a basic introduction to modular arithmetic, where Modulo 2 arithmetic is the type that deals exclusively with zeroes and ones. Although not mentioned in this story, important considerations other than speed in doing calculations include things like precision and accuracy.

Meanwhile, if you're wondering about the illustration for this section, we generated it using Stable Diffusion's 2.1 Demo, using the prompt "artificial intelligence finds a faster way to multiply matrices". If you think about it, it's an AI telling us what another AI looks like!...

Finding the Limits of Math That Can Be Done

The limitations of the AI-developed matrix multiplication algorithm point toward another of the big challenges mathematicians took on in 2022. Specifically, the challenges associated with determining when calculations can be done with confidence they'll deliver good results and identifying the limits where they stop working well.

The biggest math story in this category involves a new proof that also relied upon advanced computing techniques to find out when and under what conditions Euler's equations describing fluid mechanics might "blow up".

In this case, "blowing up" means the math breaks down and starts providing results that are violently unstable. That's similar to the situation where you have a simple fraction with zero in the denominator, the result for which is undefined when you attempt to perform the implied division operation. But as you're about to see, that's not the only challenge associated with the proof that's been advanced:

In a preprint posted online last month, a pair of mathematicians has shown that a particular version of the Euler equations does indeed sometimes fail. The proof marks a major breakthrough — and while it doesn’t completely solve the problem for the more general version of the equations, it offers hope that such a solution is finally within reach. “It’s an amazing result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”

There’s just one catch.

The 177-page proof — the result of a decade-long research program — makes significant use of computers. This arguably makes it difficult for other mathematicians to verify it. (In fact, they are still in the process of doing so, though many experts believe the new work will turn out to be correct.) It also forces them to reckon with philosophical questions about what a “proof” is, and what it will mean if the only viable way to solve such important questions going forward is with the help of computers.

You know you're on the bleeding edge of progress in maths when new questions like these are being raised!

Should the new proof hold, it marks a major development toward determining if the more general Navier-Stokes fluid dynamics equation could be similarly vulnerable to blowing up as well. The challenge of answering that question is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems, where the mathematicians who definitively determine if there are or if their are not any conditions that could cause the equation to blow up will win a $1 million prize. There's been a lot of mathematicians nibbling around the edges for resolving that question, but with no results confirmed as yet.

Approximate and Exact Solutions

When we write up the biggest math story of the year, we're really seeking out the biggest math story that offers real practical application. During the year, we featured a story with that headline about such an application that involved finding a simple and relatively accurate approximation to math that might otherwise require more significant computing resources to obtain results.

Here, several chemists realized that the calculus integral developed by epidemiologial pioneers W. O. Kermack and A. G. McKendrick to describe how a contagious disease might propagate through a population was identical to the math used to describe the progress of an autocatalytic reaction in chemistry. That integral has no direct solution, but it happened to be math for which they developed an approximate algebraic formulation to solve problems involving autocatalytic reactions that they realized could be applied to quickly solve the Kermack-McKendrick integral with a small margin of error.

One of the authors described how they found the connection:

Dr. Baird presented the model in May at the Southeastern Theoretical Chemistry Association meeting in Atlanta.

"The World Health Organization could program our equation into a hand-held computer," Dr. Baird says. "Our formula is able to predict the time required for the number of infected individuals to achieve its maximum. In the chemical analog, this is known as the induction time."

The formula is capable of predicting the number of hospitalizations, death rates, community exposure rates and related variables. It also calculates the populations of susceptible, infectious and recovered individuals, and predicts a clean separation between the period of onset of the disease and the period of subsidence....

"The rate of infection initially accelerates until it reaches a point where the infection rate is balanced by the recovery rate of infected individuals, at which point the number of infected people peaks and then starts to decay," he says.

That mechanism reminded him of the mechanism that governs an autocatalytic reaction.

"I subsequently learned that the mathematical description of the spread of infectious diseases was first described by Kermack and McKendrick," Dr. Baird says.

"When I read their paper, I realized that their mechanism was exactly the same as that of an autocatalytic reaction, where a catalyst molecule combines with a reactant molecule to produce two catalyst molecules," he says. "The rate of production of catalyst molecules accelerates until it is balanced by the rate of decay of the catalyst to form the product."

This story is also about the tradeoff between accepting small errors in return for obtaining reasonably accurate results with speed. But what if you could develop a formula that provided an exact solution?

That scenario is playing out at the University of Bristol, where the math needed to realistically modeling the process of diffusion has been developed by Toby Kay, an engineering mathematics PhD student, and Dr. Luca Giuggioli, who has been working on the math of diffusion for some time.

A groundbreaking mathematical equation that could transform medical procedures, natural gas extraction, and plastic packaging production in the future has been discovered.

The new equation, developed by scientists at the University of Bristol, indicates that diffusive movement through permeable material can be modeled exactly for the very first time. It comes a century after world-leading physicists Albert Einstein and Marian von Smoluchowski derived the first diffusion equation, and marks important progress in representing motion for a wide range of entities from microscopic particles and natural organisms to man-made devices.

Until now, scientists looking at particle motion through porous materials, such as biological tissues, polymers, various rocks and sponges have had to rely on approximations or incomplete perspectives.

The findings, published today in the journal Physical Review Research, provide a novel technique that presents exciting opportunities in a diverse range of settings including health, energy, and the food industry.

Here's a sampling of where the new diffusion math described in their paper might be applied in its next steps:

Further research is needed to apply this mathematical tool to experimental applications, which could improve products and services. For example, being able to model accurately the diffusion of water molecules through biological tissue will advance the interpretation of diffusion-weighted MRI (Magnetic Resonance Imaging) readings. It could also offer more accurate representation of air spreading through food packaging materials, helping to determine shelf life and contamination risk. In addition, quantifying the behavior of foraging animals interacting with macroscopic barriers, such as fences and roads, could provide better predictions on the consequence of climate change for conservation purposes.

That's a lot of exciting potential, but doesn't qualify as the Biggest Math Story of 2022. For that, you'll need to read on....

Persistence and Dedication

In assembling the Biggest Math Story of the year, we often discover themes or trends that make the year stand out from others. In 2018, it was the role of amateurs in advancing mathematical knowledge. In 2019, social media proved to be an enabling factor in many of the year's bigger math stories. 2020 was completely defined by the failure of the world's premier epidemiological models to forecast the coronavirus pandemic. And we've already mentioned 2021 as the year artificial intelligence contributed to significant mathematical discoveries.

We think the underlying theme for 2022 is persistence and dedication. Like 2018, amateurs made some very practical advances, including high school students like Glenn Bruda, who identified an improved method for integrating complex equations and Daniel Larsen, who developed a proof for a theorem about the distribution of Carmichael numbers, also known as "pseudoprime" numbers, which parallels work being done by well established mathematicians to crack number theory's "twin prime" conjecture. When you dig into these stories, these achievements are the result of these amateur mathematicians' unusual persistence and dedication.

But it's not just amateurs. We found the story of how mathematicians Eric Larson and Isabel Vogt solved a several hundred year-old conjecture about algebraic curves to be unusually charming. Here's an excerpt from Quanta Magazine's Jordana Cepelwicz' story on their discovery:

... in a proof posted online earlier this year, two young mathematicians at Brown University, Eric Larson and Isabel Vogt, have finally dealt the problem its final blow, solving it completely and systematically. The paper marks the culmination of nearly a decade of work, during which they gradually chipped away at the question, solved important related problems about what curves look like and how they behave — and also got married.

“It’s really a remarkable story,” said Sam Payne, a mathematician at the University of Texas, Austin, “for [people] that young and that early in their mathematical development to latch on to such a deep, hard problem, and then to be so persistent.”

The best part of the story is its revelation that Larson and Vogt keep chalkboards in their home so they can work on problems. If you know anything about mathematicians and their love of chalk, you appreciate their dedication!

Then again, they are professional mathematicians! Pure mathematics, as done by professional mathematicians, often doesn't directly connect to practical applications. But when it does, the outcome can be stunning.

That's the category into which we would place the work that mathematicians Jinyoung Park and Huy Tuan Pham did to prove the Kahn-Kalai Conjecture, which had been a major open problem in the field of probabilistic combinatorics. The press release announcing their proof describes how it connects to practical considerations:

The conjecture concerns determining the precise point (e.g. temperature, pressure, probability, etc.) at which a "phase transition" occurs in a large variety of systems. The systems are studied widely in statistical mechanics and graph theory. While this point is extremely hard to compute, in 2006 Jeff Kahn and Gil Kalai, past IAS Member (1995, 2000) and frequent visitor, conjectured that it is very close to another parameter which is much easier to compute. If true, it could be possible to approximate well when phase transitions occur. This has been called “the expectation threshold conjecture.’’

That's the sort of thing that would be very useful in applications like directing chemical reactions, managing an electrical grid, or determining the resilience of a financial system. What's more remarkable is that Park and Pham's proof is just a mere six pages long. If you're interested in finding out more, the invaluable Quanta Magazine introduces more information about their proof.

There is one more story that needs to be told under 2022's unofficial theme of persistence and dedication. That story is graduate student Jared Duker Lichtman's proof of a conjecture proposed by the prolific mathematician Paul Erdős regarding prime numbers and primitive sets. Fortunately for us, Lichtman discussed his proof with Numberphile's Brady Haran in the following video:

We've featured many Numberphile videos over the years, this one has quickly become one of our favorites. It's not the biggest math story of the year, but it's well worth your time in watching because it explores why the challenge of proving math conjectures can command such persistence and dedication!

Real Numbers Alone Cannot Describe Reality!

When we wrapped up the biggest stories in math for 2021, the publication of the paper by Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascués demonstration that a theory of quantum mechanics based only upon real numbers is false just barely missed our cutoff date for inclusion.

This is a massively important story with universal impact, because it means the universe itself cannot exist without complex numbers, or rather, a combination of both real numbers and imaginary numbers!

That's not just hyperbole coming from mathematicians. That's the direct outcome from two experiments that put real numbers to the test of explaining physical reality and found them wanting. Here's some background for the experiments:

Some physicists have attempted to build quantum theory using real numbers only, avoiding the imaginary realm with versions called “real quantum mechanics.” But without an experimental test of such theories, the question remained whether imaginary numbers were truly necessary in quantum physics, or just a useful computational tool.

A type of experiment known as a Bell test resolved a different quantum quandary, proving that quantum mechanics really requires strange quantum linkages between particles called entanglement. “We started thinking about whether an experiment of this sort could also refute real quantum mechanics,” says theoretical physicist Miguel Navascués of the Institute for Quantum Optics and Quantum Information Vienna. He and colleagues laid out a plan for an experiment in a paper posted online at arXiv.org in January 2021 and published December 15 in Nature.

In this plan, researchers would send pairs of entangled particles from two different sources to three different people, named according to conventional physics lingo as Alice, Bob and Charlie. Alice receives one particle, and can measure it using various settings that she chooses. Charlie does the same. Bob receives two particles and performs a special type of measurement to entangle the particles that Alice and Charlie receive. A real quantum theory, with no imaginary numbers, would predict different results than standard quantum physics, allowing the experiment to distinguish which one is correct.

Fan and colleagues performed such an experiment using photons, or particles of light, they report in a paper to be published in Physical Review Letters. By studying how Alice, Charlie and Bob’s results compare across many measurements, Fan, Navascués and colleagues show that the data could be described only by a quantum theory with complex numbers.

That wasn't the only experiment to suggest it takes both real and imaginary numbers to describe reality.

Another team of physicists conducted an experiment based on the same concept using a quantum computer made with superconductors, materials which conduct electricity without resistance. Those researchers, too, found that quantum physics requires complex numbers, they report in another paper to be published in Physical Review Letters. “We are curious about why complex numbers are necessary and play a fundamental role in quantum mechanics,” says quantum physicist Chao-Yang Lu of the University of Science and Technology of China in Hefei, a coauthor of the study.

The math of complex numbers is essential to the modern world, and is increasingly so with the development of advanced electronics and other technologies that are being built in the microscopic scales where quantum mechanics define what's possible. That central role makes the story of how real and imaginary numbers are essential for the universe's existence the Biggest Math Story of 2022. There just aren't any practical applications that are bigger than that!

Bonus Update: Physicist Sabine Hossenfelder discussed whether complex numbers exist back in March 2021, back when the paper was just a preprint! Here's the video:

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions and our coverage of other math stories during 2022:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Biggest Math Story of 2019
• The Biggest Math Story of 2020
• The Biggest Math Story of 2021
• The Biggest Math Story of 2022
• The Invention of Imaginary Numbers
• The 37 Percent Solution
• The Oddness of Odd Numbers
• An Introduction to the Riemann Hypothesis
• Wordle for Maths
• How to Use a Measuring Tape to Assess Your Health Risk
• An Improved Method for Integrating Complex Equations
• The Binary Code Hidden Inside a Trick for Multiplying Numbers
• The Bungled Math Behind Biden's Failure to Approve Offshore Oil Permits
• The Math Inside the Magic
• How Long to Crack Your Password with Brute Force?
• Homer Simpson Is Made of Maths
• Words for Describing Probabilities
• Maxwell's Equations
• The Lurking Chaos in Ecosystems
• Are You Waking Up on the Right Side of the Bed?
• Maths Prove Slowly Rotating Black Holes Are Stable
• Can Statistics Catch a Cheater at Chess?
• The Dumbest Way to Solve a Maze
• A Mathematical Intersection Between Epidemiology and Chemistry
• What to Get a Maths Enthusiast for Christmas

This is Political Calculations' final post for 2022. Thank you for passing time with us this year - we hope you have a wonderful holiday season! We'll see you again in the New Year, which we'll start with another annual tradition by presenting a tool to help you find out what your paycheck will look like in 2023 after an increasingly cash-strapped U.S. government takes its cut from it....

Before we go, Quanta Magazine has put together an article and video with their take on the year's top three math breakthroughs, two of which will hopefully be familiar to you....

We'll see you in the new year!

From time to time, we like to offer gift advice for what to get that one person you know who's almost impossible to find a gift for. This year, we're taking on the challenge of what to get that person if they're really into maths!

Sure, you could get them books, but the problem with that is they'll either be much too advanced or much too simple for the person for whom you're gift shopping. Partly that's because math concepts, by their very nature, go to extremes. There's not much in the middle, so your better bet is to get them maths stuff.

Maths stuff can be as complicated or as simple as you can find and it will probably be very appreciated by your maths-loving gift recipient. We're going to suggest the sweet spot is represented by the seeming impossibility of a Klein Bottle.

What makes a Klein Bottle special is how it incorporates the maths of topology in its construction. It's a surface that has only one side and does not have any edges, so it neatly introduces the concept of infinity in the form of something that can also be used to store stuff, which makes it unexpectedly practical.

You can even get one on Amazon, handmade in glass by UC-Berkeley's topologist Cliff Stoll. But if you're reading this, you'll want to act quickly because he only makes a limited quantity each year!

Speaking of which, the following 10-minute Numberphile video shows how he and Lucas Clarke made them this year from borosilicate glass tubes.

That leave the challenge of how to fill a Klein Bottle, for which there is another Numberphile video, also featuring Cliff Stoll, where you can get the answer while going on an unexpected ride through topological theory and physics.