Kashiwara Kuniki, a Japanese mathematician, won the Abel Prize of the year. This aims to be equivalent to the Nobel Prize in mathematics. Dr. Kashiwara's highly abstract work combines algebra, geometry and differential equations in a surprising way.
The Norwegian Academy of Sciences and Letters, which administer the Abel Prize, announced their honors Wednesday morning.
“First of all, he resolved some open speculations – a difficult issue that existed,” said Helge Holden, chairman of the awards committee. “And second, he opened up new paths and connected areas that were previously unknown to be unconnected. This always surprises mathematicians.”
Mathematicians can use connections between different domains of mathematics to tackle counterargument problems and re-create them into concepts that better understand them.
That's why Kawakaze, 78, from Kyoto University, is “very important in many different fields of mathematics,” said Holden.
But did Dr. Kashiwara find use in his work in solving specific, real-world problems?
“No, there's nothing,” Dr. Kashiwara said in an interview.
This honor comes with 7.5 million Norwegian cloners, or about $700,000.
Unlike Nobel Prize winners, you may be surprised by the call in the middle just before the honor is publicly announced, but Dr. Kashiwara has known his honor for a week.
Norwegian Academy informs Abel Prize winners of looses similar to those used to make surprise birthday parties in spring to unsuspecting people. “My lab director told me there was a Zoom meeting at 4pm. Please attend,” Dr. Kashiwara recalled in an interview.
On video communication calls, he didn't recognize many faces. “There were a lot of non-Japanese people at the Zoom meeting and I'm wondering what's going on,” Dr. Kashiwara said.
Marit Westerguard, executive director of the Norwegian Academy, introduced himself and told Dr. Kashiwara that he was selected as Abel of the year.
“Congratulations,” she said.
Dr. Kashiwara was initially confused, having had problems with his internet connection. “I don't fully understand what you said,” he said.
When his Japanese colleagues repeated the news in Japanese, Dr. Kashiwara said:
Dr. Kashiwara, who grew up in post-war Japan, was attracted to mathematics. He recalls a common Japanese mathematical problem known as Tsurukamezan, translated as “Crane and Turtle calculations.”
The problem states: “We have a crane and a turtle. The number of heads is X and the number of legs is Y. How many cranes and turtles are there?” (For example, for 21 and 54 legs, the answer is 15 cranes and 6 turtles.)
This is a simple algebraic problem similar to what students solve in middle school. However, Dr. Kashiwara was much younger when he encountered problems and learned how to read the encyclopedia and come up with answers. “I can't remember because I was a kid, but I think I was six,” he said.
At university, he attended a seminar by Japanese mathematician Mikio Sato, and was fascinated by Sato's groundbreaking work, now known as algebraic analysis.
“Analysis, that's explained by inequality,” Dr. Kashiwara said. “Something is big or something is smaller than the others.” Algebra deals with equality and solves equations of some unknown amount. “Sato wanted to lead the world of equality into analysis.”
Real world phenomena are explained by real numbers such as 1, –4/3, and Pi. Some are known as imaginary numbers like me. This is the square root of –1 and a complex number, the sum of the actual and imagined numbers.
Real numbers are a subset of complex numbers. The real world, described by the mathematical functions of real numbers, is “surrounded by a complex world,” Dr. Kashiwara said.
For some equations with singularity (the point where the answer changes infinitely), looking at nearby behavior with complex numbers can provide insight. “So inferences from complex worlds are reflected in the singularity of the real world,” Dr. Kashiwara said.
He writes that he manually studied partial differential equations in master's paper using algebra and developed techniques he employs throughout his career.
Dr. Kashiwara's work also led to what is known as a theory of expression that helps to solve problems using knowledge of symmetry. “Imagine a figure depicted on the floor,” says Olivier Schiffman, a mathematician at the University of Paris Sacree and the National Center for Science and Research in France. “Unfortunately, it's all covered in mud and all you can see is, for example, the 15-degree sector of it.”
However, if you know that the numbers don't change when you rotate 15 degrees, you can reconstruct them by continuous rotation. Because of the symmetry, “we need to know the small parts to understand the whole,” Dr. Schiffman said. “Representational theory allows you to do that in much more complicated situations.”
Another invention of Dr. Kashiwara was called the Crystal Base. He drew inspiration from statistical physics. Statistical physics analyzes the temperatures that are important when a material changes stages, such as when it dissolves in water. The crystal base allowed us to replace complex, seemingly impossible calculations with much simpler graphs of vertices connected by lines.
“This purely combined object actually encodes a lot of information,” Dr. Schiffman said. “It opened up a whole new field of research.”
But confusingly, the crystals at the base of the crystal are completely different from the glittering facet gems that most people consider to be crystals.
“Crystal is probably not a good word,” admitted Dr. Kashiwara.
Dr. Holden said that Dr. Kashiwara's work was much more abstract than that of previous Abel Prize winners, making it difficult to explain to non-mesamatic people.
For example, last year's awardee Michel Taragland's work, studying the randomness of the universe, like the height of ocean waves, and the work of Louis Caffarelli, who was awarded two years ago, can be applied to phenomena, like the melting of ice.
Dr. Kashiwara's work is like linking some abstract ideas in mathematics to more abstract combinations, and more abstract combinations that are insightful to mathematicians working on a variety of problems.
“I don't think it's easy,” Dr. Kashiwara said. “sorry.”
Dr. Holden pointed to certain works in which Dr. Kashiwara speculated the existence of crystal-based bases as a “masterpiece of theorem,” and used 14 inductions to use inferences to recursively prove a set of claims.
“He has to solve one by solving other people, and they all connect,” Dr. Holden said. “And when a person falls, everything falls. So he can combine them in a very deep, very clever way.”
However, Dr. Holden said he could not provide a brief explanation of the evidence. “That's difficult,” he said. “You can see 14 steps.”
