24 December 2025

The Biggest Math Story of 2025

An editorial cartoon featuring an emcee opening an envelope to announce the biggest math story of the year. Image generated with Microsoft Copilot Designer.

The year that was 2025 has all but come and gone, so it's the perfect time to look back at the year's biggest math stories!

For our selection criteria, we've emphasized math stories that involve practical applications in selecting both the contenders and the Biggest Math Story of 2025, which we'll present at the end of this year's edition. That real world connection is something we find essential in exploring achievements in math, which is why we place our main focus on it.

But that doesn't stop us from appreciating the year's "pure" math accomplishments. For example, we're fans of the work by the volunteers behind the Great Internet Mersenne Prime Search project, which is dedicated to identifying these special prime numbers. Last year, they made news when a volunteer found the current record holder for the largest known prime number, M(136,279,841), the 52nd Mersenne prime.

While they didn't find a bigger prime number this year, their work to verify that no other Mersenne primes fit in-between the gaps of their previous discoveries continued. In 2025, they expanded their test results to verify that they had identified all Mersenne primes smaller than M(77,232,917), the 50th Mersenne prime. We anticipate they'll clear the distance between that prime and M(82,589,933), the 51st Mersenne prime, sometime in 2026.

There is however an enormous gap between the 51st Mersenne prime and the current-52nd Mersenne prime that may very well contain one or two additional Mersenne primes. We are very much looking forward to what they find.

That's enough gushing about prime numbers, so let's get to the biggest math stories of the year that was. Starting with....

The Elephant in the Room

2025 was a year in which avoiding any mention of developments related to Artificial Intelligence (AI) models and the Large Language Models that have come to define the machine learning technology was all but impossible. Hundreds of billions of dollars all around the world are being invested to build and develop the infrastructure needed to support advanced AI systems.

Many disciplines felt the impact of AI technology in 2025 and mathematics is one of many in which AI systems are starting to have a profound impact.

One of the more unique stories we encountered is about one of the first valid mathematical proofs ever produced by a generative AI system. That story came to our attention through a post at X and it's probably best to let that post tell the tale:

GPT-5 just casually did new mathematics.

Sebastien Bubeck gave it an open problem from convex optimization, something humans had only partially solved. GPT-5-Pro sat down, reasoned for 17 minutes, and produced a correct proof improving the known bound from 1/L all the way to 1.5/L.

This wasn’t in the paper. It wasn’t online. It wasn’t memorized. It was new math. Verified by Bubeck himself.

Humans later closed the gap at 1.75/L, but GPT-5 independently advanced the frontier. A machine just contributed original research-level mathematics.

If you’re not completely stunned by this, you’re not paying attention.

We’ve officially entered the era where AI isn’t just learning math, it’s creating it.

Here's the X post in which Bubeck announced the accomplishment.

But as Bubeck later notes, while GPT-5 did something that was both new and novel, humans still beat AI to the punch and delivered a better proof focused on the convex optimization problem that outperforms the AI-generated proof:

Now the only reason why I won't post this as an arxiv note, is that the humans actually beat gpt-5 to the punch :-). Namely the arxiv paper has a v2 https://arxiv.org/pdf/2503.10138v2 with an additional author and they closed the gap completely, showing that 1.75/L is the tight bound.

While that better proof clearly beat its accomplishment, the unexpected artificially generated proof demonstrates AI systems are becoming more capable and useful. Because they are, more things are becoming possible.

Later in the year, a paper revealed another major advancement for the use of AI technologies in maths. Here's that story, which involves events that took place in 2024:

At the 2024 International Mathematical Olympiad (IMO), one competitor did so well that it would have been awarded the Silver Prize, except for one thing: it was an AI system. This was the first time AI had achieved a medal-level performance in the competition's history....

The AI is AlphaProof, a sophisticated program developed by Google DeepMind that learns to solve complex mathematical problems. The achievement at the IMO was impressive enough, but what really makes AlphaProof special is its ability to find and correct errors. While large language models (LLMs) can solve math problems, they often can't guarantee the accuracy of their solutions. There may be hidden flaws in their reasoning.

AlphaProof is different because its answers are always 100% correct. That's because it uses a specialized software environment called Lean (originally developed by Microsoft Research) that acts like a strict teacher verifying every logical step. This means the computer itself verifies answers, so its conclusions are trustworthy....

In addition to solving seemingly intractable math problems, AlphaProof could also be employed by mathematicians to correct their work and help them develop new theories.

The integration of a well-established proof assistant in the AI system represents a major advance and directly addresses one of generative AI technology's main shortcomings: these systems can "hallucinate" and produce absolutely garbage results. Kind of like an editorial cartoon featuring an emcee about to announce the winner of The Biggest Math Story of the Year who has too many hands and fingers.

For mathematicians though, that risk has become much smaller because of the strict discipline imposed by the proof assistant. If the attempted results fail to pass muster according to the logic imposed by the proof assistant, they're rejected.

This is probably the last year in which it might be possible to contain AI-related developments within a single section of a year-end wrap up. Fortunately for humans, none of these stories represent the Biggest Math Story of 2025!

Cracking How a Cryptocurrency Collapsed

In 2022, the cryptocurrency Terracoin collapsed. At the time, there were indications it had been the result of a coordinated effort to crash the cryptocurrency by a handful of traders who stood to profit as other investors got wiped out. But the question of exactly how it was done has been a mystery. Three years later, intrepid researchers using advanced mathematical tools and specialized software they developed cracked the mystery of how it was done. Here's an excerpt:

In a new study published in ACM Transactions on the Web, researchers from Queen Mary University of London have unveiled the intricate mechanisms behind one of the most dramatic collapses in the cryptocurrency world: the downfall of the TerraUSD stablecoin and its associated currency, LUNA. Using advanced mathematical techniques and cutting-edge software, the team has identified suspicious trading patterns that suggest a coordinated attack on the ecosystem, leading to a catastrophic loss of $3.5 billion in value virtually overnight.

The study, led by Dr. Richard Clegg and his team, employs temporal multilayer graph analysis—a sophisticated method for examining complex, interconnected systems over time. This approach allowed the researchers to map the relationships between different cryptocurrencies traded on the Ethereum blockchain, revealing how the TerraUSD stablecoin was destabilized by a series of deliberate, large-scale trades.

Stablecoins like TerraUSD are designed to maintain a steady value, typically pegged to a fiat currency like the US dollar. However, in May 2022, TerraUSD and its sister currency, LUNA, experienced a catastrophic collapse. Dr. Clegg's research sheds light on how this happened, uncovering evidence of a coordinated attack by traders who were betting against the system, a practice known as "shorting."

"What we found was extraordinary," says Dr. Clegg. "On the days leading up to the collapse, we observed highly unnatural trading patterns. Instead of the usual spread of transactions across hundreds of traders, we saw a handful of individuals controlling almost the entire market. These patterns are smoking gun evidence of a deliberate attempt to destabilize the system."

From a trading standpoint, the apparent conspiracy to crash the TerraUSD stablecoin was possible because it was very thinly traded and lacked the liquidity needed to offset the determined efforts by those shorting the cryptocurrency. But more importantly, Clegg's team identified and developed the mathematical tools needed to detect such an effort, which will have real world application as digital currencies like stablecoins become more common.

"These kids today, with their loud music and hula hoops!..."

Every year, Ig Nobel Prizes are awarded to unusual scientific research that, given the subjects involved, make people laugh and then make them think. Occasionally, there's a mathematics category. Here's our contender for next year's Ig Nobel Prizes, which features the first ever results that "explain the physics and mathematics of hula hooping":

"We were specifically interested in what kinds of body motions and shapes could successfully hold the hoop up and what physical requirements and restrictions are involved," explains Leif Ristroph, an associate professor at New York University's Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the Proceedings of the National Academy of Sciences....

The results showed that the exact form of the gyration motion or the cross-section shape of the body (circle versus ellipse) wasn't a factor in hula hooping.

"In all cases, good twirling motions of the hoop around the body could be set up without any special effort," Ristroph explains.

However, keeping a hoop elevated against gravity for a significant period of time was more difficult, requiring a special "body type"—one with a sloping surface as "hips" to provide the proper angle for pushing up the hoop and a curvy form as a "waist" to hold the hoop in place....

The paper's authors conducted mathematical modeling of these dynamics to derive formulas that explained the results—calculations that could be used for other purposes.

"We were surprised that an activity as popular, fun, and healthy as hula hooping wasn't understood even at a basic physics level," says Ristroph.

"As we made progress on the research, we realized that the math and physics involved are very subtle, and the knowledge gained could be useful in inspiring engineering innovations, harvesting energy from vibrations, and improving robotic positioners and movers used in industrial processing and manufacturing."

Makes you laugh, then makes you think. While this hula hooping story may be a perfect contender for the Ig Nobel Prizes, it's not the Biggest Math Story of 2025. For that story, please scroll down to the next section....

Unifying the Math for Describing How Fluids Behave at All Scales of the Physical Universe

In 1900, David Hilbert identified what he thought would be the 23 biggest challenges mathematics had to offer. 125 years later, over half of those challenges are still waiting to be fully resolved but Hilbert's sixth challenge was very likely cracked in 2025.

That challenge lay in defining a single set of mathematical axioms that could unite the different maths needed to describe how fluid particles behave at different scales. At the smallest "microscopic" level, the motion of individual fluid particles can be described by Newton's laws of motion. Scaling up to the medium "mesascopic" level, how groups of fluid particles behave can be described with the Boltzmann equation. Scaling up once again to the largest "macroscopic" level however takes yet another formulation, with Navier-Stokes equations taking over the work of describing how the fluid itself moves.

Since 1900, a good amount progress toward resolving Hilbert's sixth problem has been made by many contributors, but a major gap remained. In March 2025, three mathematicians, Yu Deng, Zaher Hani, and Xiao Ma posted a preprint paper that filled the remaining gap in the efforts over the past 125 years to create a unified set of math to describe fluid motion at all levels. Quanta Magazine covers the achievement in uniting the math describing the quantum-level (microscopic), medium-level (mesascopic) and large (macroscopic)-scale dynamics of how fluid particles interact in the first third of the following 20-minute video that provides an excellent overview of we think is the biggest math story of the year:

We have more background in our earlier coverage of the accomplishment. At this writing, their work is still awaiting verification, but if Deng, Hani and Ma's work holds, successfully connecting the math of Newton, Boltzmann, Navier, and Stokes across these three scales of reality represents a massive achievement in mathematical theory.

If you watch the full video, Hong Wang and Joshua Zahl's proof of the three-dimensional Kakeya Conjecture (or Needle Problem), a challenge dating back more than a century that revolves around how to determine what the smallest volume is in which a needle can be spun around all possible orientations within a space. Wang and Zahl's remarkable accomplishment is a runner up for our criteria of practical application given its links to geometric measure theory and harmonic analysis.

Ultimately, the question of which achievement would claim the title of being the Biggest Math Story of 2025 came down to recognizing Deng, Hani and Ma's proof represents the culmination of a challenge set out 125 years ago to firmly underpin the physics of how fluids behave with math that works at all scales of the physical universe. It's hard to get much bigger than that in 2025.

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 2024:

This is Political Calculations' final post for 2025. Thank you visiting, we hope found the stories and analysis we've presented throughout the year to be either thought-provoking, informative, or entertaining. We'll see you again in the New Year, which we'll kick off with our annual tradition of presenting a tool to help you find out what your paycheck will look like in 2026.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon featuring an emcee opening an envelope to announce the biggest math story of the year".

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