Masaki Kashiwara, a Japanese mathematician, received this year’s Abel Prize, which aspires to be the equivalent of the Nobel Prize in math. Dr. Kashiwara’s highly abstract work combined algebra, geometry and differential equations in surprising ways.

The Norwegian Academy of Science and Letters, which manages the Abel Prize, announced the honor on Wednesday morning.

“First of all, he has solved some open conjectures — hard problems that have been around,” said Helge Holden, chairman of the prize committee. “And second, he has opened new avenues, connecting areas that were not known to be connected before. This is something that always surprises mathematicians.”

Mathematicians use connections between different areas of math to tackle recalcitrant problems, allowing them to recast those problems into concepts they better understand.

That has made Dr. Kashiwara, 78, of Kyoto University, “very important in many different areas of mathematics,” Dr. Holden said.

But have uses been found for Dr. Kashiwara’s work in solving concrete, real-world problems?

“No, nothing,” Dr. Kashiwara said in an interview.

The honor is accompanied by 7.5 million Norwegian kroner, or about $700,000.

Unlike Nobel Prize laureates, who are frequently surprised with middle-of-the-night phone calls just before the honors are publicly announced, Dr. Kashiwara has known of his honor for a week.

The Norwegian academy informs Abel Prize recipients with ruses similar to those used to spring a surprise birthday party on an unsuspecting person. “The director of my institute told me that there is a Zoom meeting at 4 o’clock in the afternoon, and please attend,” Dr. Kashiwara recalled in an interview.

On the video teleconference call, he did not recognize many of the faces. “There were many non-Japanese people in the Zoom meeting, and I’m wondering what’s going on,” Dr. Kashiwara said.

Marit Westergaard, secretary general of the Norwegian academy, introduced herself and told Dr. Kashiwara that he had been chosen for the year’s Abel.

“Congratulations,” she said.

Dr. Kashiwara, who was having trouble with his internet connection, was initially confused. “I don’t completely understand what you said,” he said.

When his Japanese colleagues repeated the news in Japanese, Dr. Kashiwara said: “That is not what I expected at all. I’m very surprised and honored.”

Growing up in Japan in the postwar years, Dr. Kashiwara was drawn to math. He recalled a common Japanese math problem known as tsurukamezan, which translates as the “crane and turtle calculation.”

The problem states: “There are cranes and turtles. The count of heads is X and the count of legs is Y. How many cranes and turtles are there?” (For example, for 21 heads and 54 legs, the answer is 15 cranes and six turtles.)

This is a simple algebra word problem similar to what students solve in middle school. But Dr. Kashiwara was much younger when he encountered the problem and read an encyclopedia to learn how to come up with the answer. “I was a kid, so I can’t remember, but I think I was 6 years old,” he said.

In college, he attended a seminar by Mikio Sato, a Japanese mathematician, and was fascinated by Sato’s groundbreaking work in what is now known as algebraic analysis.

“Analysis, that is described by the inequality,” Dr. Kashiwara said. “Something is bigger or something is smaller than the other.” Algebra deals with equalities, solving equations for some unknown quantity. “Sato wanted to bring the equality world into analysis.”

Phenomena in the real world are described by real numbers like 1, –4/3 and pi. There are also what are known as imaginary numbers like i, which is the square root of –1, and complex numbers, which are sums of real and imaginary 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” involving functions of complex numbers, Dr. Kashiwara said.

For some equations with singularities — points where the answers turn into infinity — looking at the nearby behavior with complex numbers can sometimes provide insight. “So the inference from the complex world is reflected to the singularities in the real world,” Dr. Kashiwara said.

He wrote — by hand, in Japanese — a master’s thesis using algebra to study partial differential equations, developing techniques that he would employ throughout his career.

Dr. Kashiwara’s work also pulled in what is known as representation theory, which uses knowledge of symmetries to help solve a problem. “Imagine you have a figure drawn on the floor,” said Olivier Schiffmann, a mathematician at the University of Paris-Saclay and the French National Center for Scientific Research. “Unfortunately, it is all covered in mud and all you can see is, say, a 15-degree sector of it.”

But if one knows that the figure remains unchanged when rotated by 15 degrees, one can reconstruct it through successive rotations. Because of the symmetry, “I only need to know a small part in order to understand the whole,” Dr. Schiffman said. “Representation theory allows you to do that in much more complex situations.”

Another invention of Dr. Kashiwara’s was called crystal bases. He drew inspiration from statistical physics, which analyzes critical temperatures when materials change phases, like when ice melts to water. The crystal bases allowed complex, seemingly impossible calculations to be replaced with much simpler graphs of vertices connected by lines.

“This purely combinatorial object in fact encodes a lot of information,” Dr. Schiffmann said. “It opened up a whole new area of research.”

Confusingly, however, the crystals of crystal bases are completely different from the sparkly faceted gemstones that most people think of as crystals.

“Perhaps crystal is not a good word,” Dr. Kashiwara admitted.

Dr. Holden said Dr. Kashiwara’s work was difficult to explain to non-mathematicians, because it was much more abstract than that of some earlier Abel prize laureates.

For example, the research of Michel Talagrand, last year’s laureate, studied randomness in the universe like the heights of ocean waves, and the work of Luis Caffarelli, who was honored two years ago, can be applied to phenomena like the melting of a piece of ice.

Dr. Kashiwara’s work is more like tying together several abstract ideas of mathematics into more abstract combinations that are insightful to mathematicians tackling a variety of problems.

“I think it’s not easy,” Dr. Kashiwara said. “I’m sorry.”

Dr. Holden pointed to one particular work, in which Dr. Kashiwara deduced the existence of crystal bases, as a “masterpiece of a theorem,” with 14 steps of induction, using inference to recursively prove a series of assertions.

“He has to solve one by solving the others, and they’re all connected,” Dr. Holden said. “And if one falls, the whole thing falls. So he is able to combine them in a very deep and very clever way.”

But Dr. Holden said he could not provide a simple explanation of the proof. “That’s hard,” he said. “I can see the 14 steps.”

Last year, when Andre Rouhani and Gabriela Reyes toured Culdesac Tempe, a rental development outside of Phoenix, the place looked pretty sweet. It had winsome walkways, boutique shops and low-slung white stucco buildings clustered around shaded courtyards.

The only surprise came when Mr. Rouhani, 33, a doctoral student at Arizona State University, asked about resident parking and was told there was none.

The couple had two dogs, a toddler and another baby on the way. “Long story short, we decided that all the pros outweigh the cons,” Mr. Rouhani said in a recent phone interview. The family gave its car to Ms. Reyes’ father and moved into Culdesac in December. “We do really, really love it here,” Mr. Rouhani said. “It’s the best place I’ve ever lived.”

50 States, 50 Fixes is a series about local solutions to environmental problems. More to come this year.

Modeled on towns in Italy and Greece built long before the advent of cars, Culdesac Tempe is what its developers call the country’s first neighborhood purposely built to be car free.

Ryan Johnson, the Culdesac chief executive, said he wanted to offer a blueprint for living in a walkable place, even in a state that’s car-centric and often broiling.

“It’s one of the best things we can do for climate, health, happiness, low cost of living, even low cost of government,” said Mr. Johnson, who lives at Culdesac, too. “It’s also a better lifestyle. We all become the worst versions of ourselves behind the wheel.”

While there’s a short-term parking lot for deliveries, retailers and guests, Culdesac residents are expected to get around by the nearby light rail system, as well as on buses, scooters, electric bikes and by using ride shares. There are 22 retail shops, several of them live-work spaces, and a small Korean market. So far, 288 apartment units have been built on eight of the site’s 17 acres with another 450 units planned.

There are other car-free places in the United States, mostly island getaways where people walk, bike or tool around on golf carts. But zoning requirements in most cities usually require new developments to provide residents with a minimum number of parking spots, including in the Phoenix area, a paragon of urban sprawl. Culdesac Tempe’s developers were given a special exemption from parking requirements by the City of Tempe.

“This is completely different than our modern, conventional approach to development,” said Edward Erfurt, chief technical adviser at Strong Towns, a North American nonprofit group that promotes community resilience. “We’ve just had this experiment for the last eight decades where we’ve opted to prioritize an isolated transportation system versus our natural way of working together as humans.”

Culdesac Tempe broke that mold, Mr. Erfurt said, adding, “This is a very big deal.”

Culdesac’s two- and three-story buildings are designed for the desert climate, painted bright white to reflect heat. Not having to factor in residential parking allowed its architects to configure buildings to maximize shade and to design narrow pathways that encouraged breezes and social engagement.

“The pedestrian is really the primary person, the figure that you’re developing for,” said Alexandra Vondeling, the lead architect on the project. Big expanses of glass were eschewed, awnings added over sun-facing windows, and native plants and trees put in for cooling shade. There’s a wide walkway that can accommodate emergency vehicles, but no asphalt, reducing the urban heat island effect and improving conditions for the dogs that live there, too.

The apartments range from studios to three bedroom units, renting from between $1,300 to $2,800 a month, which Mr. Johnson said were market rates. Nearly 90 percent are leased.

Some residents were drawn to Culdesac because of its car-free mission, others in spite of it. There’s a contingent, size unknown, that quietly still owns cars, just parked off-site.

Sheryl Murdock, 50, a postdoctoral researcher who lives in Canada, is renting a unit because she is frequently in Tempe for work and wanted to balance the carbon emissions from all that flying.

Ashley Weiland and her husband moved in with their young child to give up the expense of having a car and ended up getting jobs at Culdesac, she at a restaurant there and he in maintenance.

Electra Hug, 24, who works for the city of Tempe and is blind, wanted to be close to public transit and have a sense of community. It’s the first time she’s lived without the assistance of family and friends. “In order to have a good time or have fun, I do not have to cross the street,” Ms. Hug said. “It’s just super unique and really just homey.”

Mr. Rouhani and Ms. Reyes borrow her father’s car once a week for errands. Otherwise they mostly ride public transit with free passes provided by Culdesac.

Living in a place where people are not zipping about in their cars means the pace is slower, with more opportunity for connection, Mr. Rouhani said. It is the kind of community, he said, where neighbors borrow a cup of sugar from each other. In the days after their daughter was born, three different families either brought a meal, dropped off cookies, or offered to go buy them groceries. “We really feel supported and loved here,” he said.

David King, who teaches urban planning at Arizona State University, said Culdesac Tempe could prompt other developers to push for exemptions from parking requirements. And Mr. Erfurt of Strong Towns said Culdesac Tempe could pave the way, as it were, for similar car-free developments to be built in places like shuttered strip malls, which could address the affordable housing crisis, lessen loneliness and bring people closer to where they work.

“We could do all that simply by decoupling parking from development,” Mr. Erfurt said. “In every market, people are looking for that.”