With 2021 just about over, it's time to celebrate the year's biggest stories in math!

Because we're of a practical bent, we seek out maths stories that involve doing math to do things, whether that's making progress toward resolving long-standing questions, developing practical applications, or otherwise impacting the real world. We've provided deeper information about the topics covered in our summary of 2021's biggest math stories where we can, so be sure to follow the links if you want to learn more.

We'll also identify what we think qualifies as the biggest math story of the year. Buckle up, it's time to get started!...

Making Connections with Sheaves

What better way to start than with the biggest pure mathematics story of 2021? Mathematicians Laurent Fargues and Peter Scholze made significant progress toward achieving one of the goals of the Langlands Program, which seeks to develop a grand unified theory of mathematics by connecting seemingly separate branches of math. Their achievement provides a means of translating the math of p-adic groups to Galois groups using sheaves, which are special tools that can connect local and global properties within a geometric or topological space.

The featured video lecture in this section will provide a basic introduction to them, but you'll have to make it through to the 51-minute mark before they're even introduced because of having to go through the foundational material it takes to get there. And then, there are at least four more video lectures that follow it to build on the concept!

Fargues and Scholze's achievement represents something of a halfway point in the Langlands Program's quest, because the challenge of starting with the math of Galois groups and going in the opposite direction to translate it into p-adic groups remains. But the tool they used to pull off their accomplishment has turned up in a more practical application as well.

Robert Ghrist and Jakob Hansen utilized sheaves to develop a new mathematical model for more realistically model how opinions are disseminated over social networks, where they also deployed the math of Laplacian operators and diffusion dynamics.

To do this, Ghrist and Hansen used topological tools called sheaves, previously used in their group. Sheaves are algebraic data structures, or collections of vector spaces, that are tethered to a network and link information to individual nodes or edges. Using a transportation network as an illustrative example, where train stations are nodes and the tracks are the edges, sheaves are used to carry information about the network, such as passenger counts or the number of on-time departures, not only for specific stations but also on the connections between stations.

"These vector spaces can have different features and dimensions, and they can encode different quantities and types of information," says Ghrist. "So the sheave consists of collections of vectors over top of each node and each edge with matrices that connect them all together. Collectively, this is a big data structure floating over top of your network."...

By incorporating Laplacians into their "discourse sheaves," the researchers were able to create an opinion dynamics model that was incredibly flexible and able to incorporate a wide variety of scenarios, parameters, and features. This includes the ability to have agents who can lie about their feelings on a specific topic or tell different opinions to others depending on how they are connected, all within a rigorous and testable mathematical framework.

Ghrist and Hansen's new approach to modeling how information spreads in a network has the potential to greatly improve the understanding of how information diffuses in societies. That in turn has the potential to improve not just polling, but also fields like public health, where it can be used to more effectively develop things like vaccination strategies or to direct scarce resources.

Or to improve pandemic modeling, which became the biggest math story of 2020 because of its failures.

Building Better Models

The failures of pandemic modeling in 2020 pointed to the need to develop better mathematical models of how things work in the real world. 2021 saw developments toward developing more effective models that are improving our understanding of reality.

One example of that happening lies in the work of Cambridge University researchers Eugene Terentjev and Neil Ibata to more accurately model how exercise builds muscles in a new paper. You might think this would be a very well-studied and established science, but until 2021, there was very little understanding for how that actually works. Here's how they built up their biophysical model that focused on the role of titin proteins in building muscle into something that could explain real world observations:

Terentjev and Ibata set out to constrict a mathematical model that could give quantitative predictions on muscle growth. They started with a simple model that kept track of titin molecules opening under force and starting the signaling cascade. They used microscopy data to determine the force-dependent probability that a titin kinase unit would open or close under force and activate a signaling molecule.

They then made the model more complex by including additional information, such as metabolic energy exchange, as well as repetition length and recovery. The model was validated using past long-term studies on muscle hypertrophy.

"Our model offers a physiological basis for the idea that muscle growth mainly occurs at 70% of the maximum load, which is the idea behind resistance training," said Terentjev. "Below that, the opening rate of titin kinase drops precipitously and precludes mechanosensitive signaling from taking place. Above that, rapid exhaustion prevents a good outcome, which our model has quantitatively predicted."

This story provides a good example of why we focus on practical applications for math. Mathematical models can do many things, but unless their results are continually compared with observations to validate them, even the most seemingly impressive math will fall far short of its potential.

That brings us to another example where better modeling is finally resolving a challenge that has existed for over a century: how to more accurately represent turbulent flow in a fluid.

It's no accident that the math of fluid dynamics makes repeated appearances when we cover the biggest math stories of a year, due to the immense practical value advancements in computational fluid dynamics can deliver. This year, it's because a team of researchers from the University of California at Santa Barbara and the University of Oslo led by Björn Birnir and Luiza Angheluta have published a paper providing a mathematical equation describing how fluids behave when they transition from laminar to turbulent flow in a boundary layer where the fluid is moving past a solid surface.

That's a challenge that some of the biggest names in physics and mathematics have been working toward for a very long time. The following excerpt introduces that history to put Birnir's and Angheluta's achievement into context:

This phenomenon was first described around 1920 by Hungarian physicist Theodore von Kármán and German physicist Ludwig Prandtl, two luminaries in fluid dynamics. "They were honing in on what's called boundary layer turbulence," said Birnir, director of the Center for Complex and Nonlinear Science. This is turbulence caused when a flow interacts with a boundary, such as the fluid's surface, a pipe wall, the surface of the Earth and so forth.

Prandtl figured out experimentally that he could divide the boundary layer into four distinct regions based on proximity to the boundary. The viscous layer forms right next to the boundary, where turbulence is damped by the thickness of the flow. Next comes a transitional buffer region, followed by the inertial region, where turbulence is most fully developed. Finally, there is the wake, where the boundary layer flow is least affected by the boundary, according to a formula by von Kármán.

The fluid flows quicker farther from the boundary, but its velocity changes in a very specific manner. Its average velocity increases in the viscous and buffer layers and then transitions to a logarithmic function in the inertial layer. This "log law," found by Prandtl and von Kármán, has perplexed researchers, who worked to understand where it came from and how to describe it...

In the 1970s, Australian mechanical engineer Albert Alan Townsend suggested that the shape of the mean velocity curve was influenced by eddies attached to the boundary. If true, it could explain the odd shape the curve takes through the different layers, as well as the physics behind the log law, Birnir said.

Fast forward to 2010, and mathematicians at the University of Illinois released a formal description of these attached eddies, including formulas. The study also described how these eddies could transfer energy away from the boundary toward the rest of the fluid. "There's a whole hierarchy of eddies," Birnir said. The smaller eddies give energy to the larger ones that reach all the way into the inertial layer, which helps explain the log law.

However, there are also detached eddies, which can travel within the fluid, and these also play an important role in boundary layer turbulence. Birnir and his co-authors focused on deriving a formal description of these. "What we showed in this paper is that you need to include these detached eddies in the theory as well in order to get the exact shape of the mean velocity curve," he said.

Their team combined all these insights to derive the mathematical formulation of the mean velocity and variation that Prandtl and von Kármán first wrote about some 100 years earlier. They then compared their formulas to computer simulations and experimental data, validating their results.

"Finally, there was a complete analytical model that explained the system," Birnir said.

Having a mathematical formula, fluid dynamicists and physicists can now more accurately model the physical flow of fluids, whether they be air over an airplane's wing, liquid mixtures passing through a pipe at a chemical plant, or weather systems passing over the Earth.

That's a big deal, but it's not the biggest math story of the year. That's coming up next....

The Year AI Made Its Mark on Maths

There wasn't one big math story in 2021 so much as there were multiple, independent math stories pointed to what the biggest math story of the year would prove to be. And that story is about the rise of Artificial Intelligence (AI) as a tool for making serious advances in multiple fields of maths.

That's on top of the stories where either "machine learning" or "neural nets" appeared as a player behind an accomplishment. What changed in 2021 was the development of AI systems to first generate and then prove mathematical conjectures.

That goes beyond the capabilities of previous knowledge-based tools, such as proof assistants, which also made a big leap in 2021.

What AI promises is a serious boost to the productivity of mathematicians. 2021 is the first year in which that promise became something more than a potential future event, making the rise of AI the biggest math story of 2021.

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, along with our coverage of other math stories during 2021:

• 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
• A Simple Introduction to Quantum Tunneling
• How Different Were Neanderthals?
• The Most Romantic Shape in Mathematics
• How Much SARS-CoV-2 Coronavirus Is There in the World?
• Inventions in Everything: Better 2D Maps of a 3D World
• A Computational Mystery for Pi Day
• Solving for Three
• The Magic of the Pigeonhole Principle
• How Lucky Is Too Lucky in Minecraft?
• COVID and the Magnitude of the Border Migration Crisis in Arizona
• The Hole at the Bottom of Math
• Counting in Cuneiform
• Say, What's this Bitcoin Thing We Keep Hearing About in the News?
• The Logistic Map and the Emergence of Complexity
• Gambling on the Martingale Strategy
• The Mathematically Defined Egg
• Inventions in Everything: Artificial Intelligence-Generated Patents
• Guessing to Reach Solutions and Newton's Fractal
• Spooky Maths
• Applied Geometry in Babylon
• Heavy Tails and Taylor's Law
• Ramanujan Machines: AI and the Future of Mathematics

This is Political Calculations final post for 2021. Thank you for passing time with us this year and have a Merry Christmas and a wonderful holiday season. We'll see you again in the New Year, which we'll kick off with another annual tradition by presenting a tool to help you find out what your paycheck will look like in 2021 after the U.S. government takes its cut from it....

Before we go, we're not the only ones paying attention to the biggest stories in maths. Quanta Magazine has put together a video with their take on the year's top math stories:

We'll see you in the new year!