The Navier–Stokes equations form the foundation of fluid dynamics, describing the motion of viscous fluids and gases. The core of the problem lies in determining whether these equations always have smooth (continuous) solutions. As highlighted by media outlets, including Hromadske’s coverage of OpenAI’s achievement, the company’s algorithms constructed an example of a finite-time blowup, thereby confirming the scenario of breakdown in smoothness.

To achieve this result, the swarm system exchanged 2.7 million messages and generated 130 billion tokens. OpenAI emphasized that it is waiving the $1 million prize from the Clay Mathematics Institute in favor of scientific progress. At the same time, as analytical publication Implicator AI highlights regarding the Clay Institute’s position, the international scientific community is in no rush to officially record a victory, as peer review and proof verification by experts will take months. Furthermore, observers at ResultSense point to a priority dispute concerning source usage and algorithmic autonomy.

Outstanding Awards and Recognition

Breakthrough Prize in Mathematics — the prestigious 2025 prize was awarded to the eminent mathematician Dennis Gaitsgory for his fundamental contributions to the Geometric Langlands Program, in particular for proving the Geometric Langlands Conjecture in characteristic zero.

New Horizons in Mathematics Prize — research in the field of conformal probability was recognized with the prize for young mathematicians, awarded to Ewain Gwynne of the University of Chicago.

Professor John Pardon also became a co-recipient of the New Horizons in Mathematics Prize for his significant results in geometry and topology.

Major Mathematical Progress

Substantial progress in solving a classical mathematical problem — in 2025, an important step was made toward resolving Hilbert’s Sixth Problem, which aims to provide a rigorous mathematical foundation for physical theories.

Mathematics and Artificial Intelligence

AI as a tool for mathematical thinking — a number of studies in 2025 demonstrated how artificial intelligence is increasingly contributing to solving complex mathematical problems and education: experimental AI systems are already approaching the gold medal standard at the International Mathematical Olympiad or achieving high scores in prestigious mathematical competitions, indicating a transformation in approaches to mathematical thinking and education.

Other Notable Results

Collective scientific achievements — large-scale collaborative projects by mathematicians worldwide, such as comprehensive collective proofs in the Langlands theory or new constructions in combinatorics and knot theory, continue to influence fundamental mathematics and its applications.

These events and achievements demonstrate that in 2025 mathematics remains one of the most dynamic and productive sciences, shaping our understanding of complex systems, algorithms, and the structure of the universe.

Main image by Dung Tran from Pixabay

Square Root Day is an unofficial holiday celebrated on the day when the day of the month and the number of the month are the same and equal the square root of the last two digits of the current year. In each century, there are only 9 such dates.

In the 21st century, Square Root Day has already been celebrated on:

• January 1, 2001 (01/01/01);
• February 2, 2004 (02/02/04);
• March 3, 2009 (03/03/09);
• April 4, 2016 (04/04/16).

The upcoming dates for this holiday will be:

• June 6, 2036 (06/06/36);
• July 7, 2049 (07/07/49);
• August 8, 2064 (08/08/64);
• September 9, 2081 (09/09/81).

The dates of Square Root Day in any century are always the same because only the last two digits of the year are used to determine the date of the holiday. For mathematical reasons, the holiday never falls on any days between October and December.

For the first time (and the only time in the 20th century), Square Root Day was celebrated on September 9, 1981 (09/09/81). The founder of this holiday is considered to be Ron Gordon, a high school teacher from Redwood City, California, USA. Enthusiasts of the holiday suggest serving root vegetables cut into cubes (symbolizing square roots) or baked goods shaped like square root symbols on this day.

In 2004, Square Root Day coincided with Groundhog Day (February 2, 2004), and in 2016 with Internet Day (April 4, 2016). The date January 1, 2001, coincided with the beginning of the 21st century and the 3rd millennium. In 2009, in honor of Square Root Day (March 3, 2009), Ron Gordon held a contest with a prize pool of $339.

It is worth noting that the 2025 Square Root Day was special, as not only were the last two digits of the year a square (25 = 5²), but the entire year itself was a perfect square (2025 = 45²). The previous “square” year was 1936 (1936 = 44²), though Square Root Day was not celebrated back then, and the next such year will be 2116 (2116 = 46²).

Although Square Root Day is not widely recognized, it showcases the beauty of mathematics, drawing attention to interesting numerical facts and their presence in everyday life. In line with Sustainable Development Goal 4: Quality Education, this expands opportunities for promoting education in mathematics and the natural sciences.

The official website of this holiday is https://squarerootday.net.