In addition to preserving fading knowledge and revisiting old questions, mathematicians and computer scientists had an exciting year of breakthrough in set theory, topology and artificial intelligence. They made new progress on fundamental questions in the field, celebrated connections between distant areas of mathematics, and saw the links between mathematics and other disciplines grow. Many results were only partial answers and some promising avenues of exploration were dead ends.
A major 40-year-old work that was in danger of being lost was finally presented in a book this fall, which topologists saw as a sign of hope. A geometric tool was created 11 years ago and has gained new life in a different mathematical context. New work in set theory made it possible to understand how many real numbers there really are. This was one of the questions that received an answer this year.
There is math in a vacuum. One of the most successful concepts in physics is quantum field theory. Computers are becoming more and more important tools for mathematicians, who use them to solve impossible problems and even verify complicated proof. This year has seen new progress in understanding how machines got so good at it as they become better at solving problems.
Grace Park was featured in a magazine.
It is tempting to think that a mathematical proof would stay around forever. A seminal result from 1981 was in danger of being lost to obscurity as the few remaining mathematicians who understood it grew older and left the field. The proof of the Poincaré conjecture showed that certain shapes that are similar in some ways to a four-dimensional sphere must also be similar in other ways. Topologists have different ways of determining when two shapes are the same. A new book called The Disc Embedding Theorem establishes the inescapable logic of the surprising approach and firmly establishes the finding in the mathematical canon.
The Smale conjecture asks if the four-dimensional sphere has all the symmetries it has. The answer is no, more kinds of symmetries exist, and in order to find them, he started a search, with new results appearing as recently as September. Two mathematicians developed a framework for combining geometry and topology, and created a new set of tools for approaching problems in those fields. In January, a column was written about the origins of topology and an explainer was written about the subject of homology.
Quanta Magazine is written by Olena Shmahalo.
Deep neural networks, a form of artificial intelligence built upon layers of artificial neurons, have become increasingly sophisticated and powerful. Traditional machine learning theory says their huge numbers of parameters should result in overfitting and an inability to generalize, but clearly something else is happening. It turns out that older and better-understood machine learning models are similar to idealized versions of neural networks, suggesting new ways to understand and take advantage of the digital black boxes.
There have been setbacks as well. There is a good chance that the artificial intelligence networks will always be 888-609- 888-609- 888-609- 888-609- 888-609- The recent work shows that the problem of gradient descent is a fundamentally difficult one, meaning some tasks may be beyond reach. A major paper describing how to create error-resistant qubits was withdrawn in March, forcing scientists to realize that such a machine may be impossible. In a column and video, Scott wrote about why quantum computers are so difficult to work with.
Quanta Magazine is written by Olena Shmahalo.
How many real numbers exist? This year saw major developments toward an answer to the question, which has been a provocative question for more than a century. David Asper and Ralf Schindler published a proof in May that combined two previously antagonistic axioms: A variation of one of them, known as Martin's maximum, implies the other. The result suggests that the number of real numbers is bigger than initially thought, which in turn means that the axioms are more likely to be true. The continuum hypothesis states that there is no size of infinite between the set of all natural numbers and the continuum of real numbers. Hugh Woodin, the original creator of (), has posted new work that suggests the continuum hypothesis is right after all.
There were other decades-old problems that were addressed by modern solutions. In 1900, David Hilbert came up with 23 significant questions, and this year saw incomplete answers to the 12th problem, about the building blocks of certain number systems, and the 13th about the solutions to seventh-degree polynomials. Multiplicative inverses actually exist in more complicated structures than mathematicians thought, when the unit conjecture is false. Alex Kontorovich explored the greatest unsolved problem in mathematics in an essay and video in January.
Matteo Bassini is a photographer.
A mathematical advance that answers a major question and provides a new avenue of exploration is often a great mathematical advance. The geometric object created by Fargues and Fontaine helped their research. The Fargues-Fontaine curve was further connected to number theory and geometry when combined with Peter Scholze's ideas about perfectoid spaces. Scholze said it was a kind of wormhole between two different worlds.
An interview with Ana Caraiani, whose work has helped strengthen and improve similar connections between disparate areas of math, and an examination of the Galois groups of symmetries at the heart of the original Langlands conjectures were included in other ruminations on the program.
Researchers use partial differential equations to describe and understand real-world systems. PDEs are difficult to solve. Two new types of neural networks, DeepONet and the Fourier neural operator, have emerged to make this work easier. The nets can map an infinite-dimensional space onto another infinite-dimensional space with the power to approximate operators. The new systems can solve existing equations faster than conventional methods, and they may also help provide PDEs for systems that were previously too complicated to model.
Computers have been helpful to mathematicians this year. In January, Quanta reported on new quantum computers that would be able to process nonlinear systems, where interactions can affect themselves, by first approximating them as simpler, linear ones. A team of mathematicians used modern hardware and software to prove that there are no more types of special tetrahedra than the ones discovered 26 years ago, and a digital proof assistant named Lean verified the correctness of an inscrutable.
Quanta Magazine is written by Olena Shmahalo.
Both physics and mathematics have always been innovative. The concept of quantum field theory has been a huge success, but it rests on shaky mathematical ground. Bringing mathematical rigor to quantum field theory would help physicists work in and expand that framework, but it would also give mathematicians a new set of tools and structures to play with. In a four-part series,Quanta examined the main issues currently getting in mathematicians' way, explored a smaller-scale success story in two dimensions, discussed the possibilities with QFT specialist Nathan Seiberg, and explained in a video the most prominent QFT of all: the Standard Model
