The Biggest Math Breakthroughs in 2022

A mathematician is like an archeologist, brushing dust off the hidden structures of the world. The structures show mathematicians are inescapable. They could not have been any different. Each year the mathematical frontier expands as new discoveries are made, but the sprawl of subdisciplines shrinks a little as connections are found between seemingly far- flung domains.

It can be hard to convey how shocking some of these connections are to the non expert. We have done our best in the years to come.

A 912-page paper showing that slowly rotating black holes are going to keep on rotating until the end of time was a proof of six pages.

There are some mathematical results that can be hard to understand. To make important proof and techniques understandable to a broader range of mathematicians, and to discover new bounds on the number of integer solutions to equations called elliptic curves, was the work of a number theorist at Duke University who was talked to byQuanta.

Alex Kontorovich talked about the Langlands program in a video and column.

Though many of these results have no immediate practical applications, it is difficult to know what abstract result will become crucial to come up with new, secure ciphers or updating error-correcting codes.

There isn't a single path to becoming a mathematician who can uncover a fundamental truth that nobody else can. Some people focus on mathematics from a young age, while others do it later in life. Some people satisfy a stereotype while others don't.

It's nice to admit that we don't know what's going on when it comes to how the human mind makes leaps of mathematical reasoning

To recognize outstanding mathematical achievement for existing work and for the promise of future achievement, up to four mathematicians younger than 40 are given a gold coin engraved with the head of Archimedes. June Huh, James Maynard, Maryna Viazovska, and Hugo Duminil-Copin received the Fields medal.

Huh said that thematics gave him the ability to search for beauty outside himself, to try to grasp something external, objective and true. His proof was recognised by the prize. He found a deep geometric structure that waslurking beneath the properties of graphs.

Maynard received an award for his discoveries. He showed that there are an infinite number of prime pairs that are not the same. If another mathematician had not shown the existence of a finite bound on the gaps between prime pairs, this would have been even more significant. It was an example of how parallel progress can be made on a problem. Maynard proved that there are an infinite number of primes that don't have a digit.

It has been known for a long time that the densest way to pack circles in a plane is in a honeycomb. Higher dimensions are not well known. The most efficient way to pack spheres is through an eight-dimensional lattice. The result was generalized to prove that the lattice minimized the energy of the system.

Duminil-Copin came up with a theory of how liquids move through porous media.

There were other prizes given this year. Dennis Sullivan came up with a new way to classify certain types of manifolds, which looked flat on a small scale but were more complicated when looked at in their entirety.

A better understanding of quasi-periodic operators, which are used to model electron behavior, was achieved by Svetlana Jitomirskaya.

The year was a good one for number theorists. A bound on the gaps between pseudoprimes can be found by a high school student and can be factored.

According to a certain measure, the largest example of a primitive set is actually primes.

Two mathematicians at the California Institute of Technology were able to prove a 1978 conjecture that predicted the sums of the form $latex efrac2in3p$. The generalized Riemann hypothesis is a hypothesis mathematicians believe to be true but have not proven. The subconvexity problem was solved.

A pair of mathematicians showed that even if there is an even or odd number of prime factors, it doesn't matter if the number is even or odd.

The group showed that no more than 1/3 of the numbers can be written as the sum of two cubes.

When d is not an odd prime, the equation can't be solved more than half the time. His hypothesis was proven.

One of the long-standing questions was shown to be correct. The 30-year-old André-Oort and 85-year-old Van der Waerden conjectures have been proved.

This year, it was proven that subsets whose reciprocals sum to 1 must be in large sets of integers. Using methods from the study of dynamical systems, it was proven that large sets of integers must have an infinite sumset.

Deep learning, the widely used artificial intelligence technique that has been used at games like chess and Go and proved very accurate at tasks like speech recognition, is being used in some areas of math. It was used to find unusual singularities in equations that model the flow of fluids. A separate team used computer-assisted proof to prove that a particular version of the equation broke down. Another group looked at the related equations to see if they could also fail. A million-dollar prize is up for grabs from the Clay Mathematics Institute.

Several groups used machine learning to solve problems in graph theory, and came up with new ideas for knot theory. Sébastien Bubeck and Mark Sellke used mathematical techniques to show how big neural networks can be.

Steve Strogatz asked Kevin Buzzard if computers can be mathematicians and Melanie Matchett Wood talked about what it takes for mathematicians to believe a result has been proven.

Not to be left behind, thegeometers had a busy year. The shape of clusters of bubbles was discovered by Emanuel and Joe in May.

There is a long-standing question about how many random points in high-dimensional space can pass through.

They were able to figure out how a circle can be cut into pieces that can be rearranged into a square. The process of figuring out whether two polygons have the same area is explained in a column.

The paper shows how to fold a polyhedron into a flat shape, as long as you allow many folds. There are more intricate relationships between the interior angles of pyramids than there are between the two-dimensional analogues. A group of mathematicians and physicists published a paper in August that showed how the wrinkling of thin materials can be caused by the curve of the material.

When Dusa McDuff and others tried to make shapes called ellipsoids into something called Hirzebruch surfaces, they found a place where they hadn't expected to find anything.

There are doubts as to whether the Kakeya conjecture is true in the domain of real numbers, but other mathematicians made progress toward proving it.

The work of Will Hide and Michael Magee, who used techniques borrowed from graph theory to show the existence of high-genus surfaces that are deeply connected with themselves, was covered byQuanta this year.

Ian Agol showed how to rank knots.

There are knots in two dimensions that can be used to show the boundaries of a surface. Even if the knots are manipulated in three-dimensional space, there are still many pairs of Seifert surfaces that are different from one another. For the first time, topologists found a pair of Seifert surfaces that were completely different from each other. Colin Adams and Lisa Piccirillo discussed their fascination with knots on an episode of the Joy of Why.

A short proof was published in March that proved the Kahn-Kalai hypothesis. In January there was a proof that it was possible to build a hypergraph in a way that met two seemingly incompatible criteria.

Graph theory results continued to come in. When graphs must inevitably become regular, or connected in such a way that every edge is connected to the same number of edges, was the question answered in April by Oliver Janzer and a team of mathematicians. The introduction to graph theory was given in a column.