Mathematicians embrace computational evidence in “Grand Unified” theory


Attempts to verify complex mathematical proofs with computers have been successful.Credits: Fadel Senna / AFP via Getty

Peter Scholze wants to start with one of its foundations and reconstruct a lot of modern mathematics. He is currently undergoing scrutiny of evidence at the heart of his search from an unlikely source of information, the computer.

Most mathematicians suspect that machines will soon replace the creative side of the profession, but some acknowledge that technology is playing an increasingly important role in research. This particular achievement can be a turning point in its acceptance.

In a lecture series in 2019 at the University of Bonn, Germany, Scholze presented an ambitious plan that he developed in collaboration with Dustin Klausen from the University of Copenhagen. did. The two researchers call it “condensed mathematics” and promise to bring new knowledge and connections between the disciplines from geometry to number theory.

Other researchers are also paying attention. Scholze is considered one of the most famous stars in math and has a proven track record of introducing innovative concepts. Emily Riehl, a mathematician at Johns Hopkins University in Baltimore, Maryland, has seen her teaching math in 50 years to graduate students if Scholze and Clausen’s vision is realized. It says. “There are many areas of math that I think his ideas will influence in the future,” she says.

In the past, much of this vision was based on technical evidence that even Scholze and Clausen were unsure whether it was correct. But earlier this month, Scholze announced a successful project to check the core of the evidence with special computer software.

Computer assistance

Mathematicians have long used computers to perform numerical calculations and manipulate complex mathematical formulas. In some cases, making a computer do a lot of iterative work has proven very successful. The most famous is the evidence that in the 1970s any map could be colored with four different colors without filling two neighboring countries with the same color. Colour.

But the system known as the Certification Assistant goes even deeper. The user enters an instruction into the system to teach the system to define a mathematical concept (object) based on a simpler object that the machine already knows. The statement can also relate to a known object, and the evidence assistant answers whether the fact is “obviously” true or false according to the current state of knowledge. If the answer is not clear, the user will have to fill in the details. Hence, the Proof Assistant forces the user to strictly formulate the logic of the arguments and inputs simpler steps that human mathematicians knowingly or unknowingly skipped.

After the hard work of turning a series of mathematical concepts into a proof assistant, the program is a library of computer code that other researchers can create to define higher-level mathematical objects. To generate. In this way, evidence assistants help humans review mathematical proofs that are time consuming to review and may even be impossible in practice.

The Evidence Assistant has long had fans, but it was his first time playing a major role at the head of the field, said Kevin Buzzard, a mathematician at Imperial College London who was part of the Scholze-Clausen collaboration. Says the result. “The big question that remains was whether we could handle complex mathematics,” says Nosri. “We have shown that they can.”

And it all happened a lot faster than anyone could have imagined. Scholze Present his challenge It was sent to an evidence assistant in December 2020 and presented by a group of volunteers led by mathematician Johan Commelin at the University of Freiburg. June 5th – within 6 months – Scholze posted on Buzzard’s blog The main part of the experiment was successful. “It’s completely insane that our interactive evidence assistants have reached a level where they can formally validate their own difficult research in a very reasonable amount of time,” Scholze wrote.

According to Scholze and Klausen, an important point in condensed mathematics is the redefinition of the concept of topology, which is one of the foundations of modern mathematics. Many of the objects that mathematicians study have topologies. Topology is a type of structure that determines which parts of an object are close together and which are not. The topology offers shape ideas, but is more adaptable than the familiar school-level geometry. The topology allows transformations that do not tear the object apart. For example, a triangle is topologically equivalent to another triangle or circle, but not a straight line.

Topology not only plays an important role in geometry, but also in functional analysis and functional research. Functions normally “exist” in space with infinite dimensions (like the wave functions on which quantum mechanics is based). It is also important for the number system called p-A base number with an exotic “fractal” topology.

Great Unity Un

Around 2018, Scholze and Klausen took a traditional approach to the concept of topology in these three mathematical universes (geometry, functional analysis, and). p– Primary – But its alternate foundation can fill in those gaps. Many of the results in each of these areas appear to be similar in other areas, although they deal with significantly different concepts. However, when the topology is “properly” defined, the similarities between the theories as instances of the same “condensed math” become apparent, the two researchers suggested. “It’s kind of a big union of the three areas,” says Clausen.

Scholze and Klausen state that they have already found a simpler “condensed” proof for many profound geometrical facts and are now able to prove previously unknown theorems. They have not yet published this.

There was a pitfall, however. To show that the geometry fits this figure, Scholze and Klausen had to prove a highly technical theorem about a set of ordinary real numbers with a linear topology. “It’s like a basic theorem that allows real numbers to be put into this new framework,” explains Commelin.

Diagram showing a network of color-coded math statements and definitions generated by certification support software.

The Proof Support Package Lean enables users to enter mathematical statements based on simpler statements and concepts already contained in the Lean library. The result seen here in the case of the main results by Scholze and Clausen is a complex network. The instructions are color-coded and grouped according to math subfields.Photo credit: Patrick Massot

Klausen remembers that he created many original ideas and worked tirelessly on the evidence until Scholze was completed “with willpower”. “That was the most amazing math achievement I’ve ever seen,” recalls Clausen. However, the debate was so complex that Scholze himself feared that there might be a small loophole that would invalidate the entire company. “It seemed convincing, but it was just too new,” says Clausen.

To validate this work, Scholze relied on Buzzard, a mathematician friend and expert on the Lean evidence-support software package. Lean was originally developed by computer scientists at Microsoft Research in Redmond, Washington, to rigorously look for errors in computer code.

Buzzard ran a multi-year program to Lean code the entire imperial math curriculum. He also tried to bring more advanced math into the system, including the concept of perfectoid space. This helped me win the 2018 Scholze Fields Medal.

Comelin, who is also a mathematician, led efforts to review Scholze and Clausen’s evidence. Commelin and Scholze decided to call the Lean project a liquid tensor experiment. respect The progressive rock band’s fluid tension experiment. Both mathematicians are fans.

The enthusiastic online collaboration continued. Over a dozen mathematicians with Lean experience took part, and the researchers were assisted by computer scientists. By early June, the team had fully leaned the core of Scholze’s evidence (the part that worried him the most). And it was all checked out – the software was able to review that part of the evidence.

Better understanding

The slim version of Scholzes proof consists of tens of thousands of lines of code, 100 times more than the original version, says Commelin. “Looking at the lean code can be very difficult to understand, especially in the current fashion.” However, researchers find that trying to build the evidence on a computer makes it easier to understand. It is also said to have been useful.

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Riehl is one of those mathematicians who have experimented with evidence assistants and teaches them in some of their bachelor’s degree programs. She says she did not use them systematically in her studies but is changing her way of thinking about the practice of creating mathematical concepts and making and proving theorems about them. “I used to think of proof and construction as two different things, but now I think they are the same thing.”

Many researchers say that machines are unlikely to be replaced by mathematicians anytime soon. The Evidence Assistant cannot read math textbooks, requires constant human input, and cannot determine whether a math statement is interesting or profound. It’s just whether it’s right, says Buzzard. Still, he added, computers could quickly point out the results of known facts that mathematicians were unaware of.

Scholze was surprised at how far the evidence assistants could go, but said he was unsure whether they would continue to play a major role in his research. “At the moment I’m not sure how you can support my creative work as a mathematician.”

Mathematicians embrace computational evidence in “Grand Unified” theory


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