David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz were awarded the Nobel Prize in Physics on Tuesday for discoveries in condensed-matter physics that have transformed the understanding of matter that assumes strange shapes. All three were born in Britain but work in the United States.

Using advanced mathematical models, the three scientists studied unusual phases, or states, of matter, such as superconductors, superfluids or thin magnetic films. Their findings have relevance for materials science and electronics.

Dr. Thouless of the University of Washington, Dr. Haldane of Princeton University and Dr. Kosterlitz of Brown University were honored by the Royal Swedish Academy of Sciences in Stockholm for “theoretical discoveries of topological phase transitions and topological phases of matter.”

Dr. Thouless was awarded half of the prize of 8 million Swedish kronor, or about $930,000, while Dr. Haldane and Dr. Kosterlitz will share the other half.

Why Did They Win?

Topology is a branch of mathematics that describes properties that change only in increments. In the early 1970s, Dr. Kosterlitz and Dr. Thouless “demonstrated that superconductivity could occur at low temperatures and also explained the mechanism, phase transition, that makes superconductivity disappear at higher temperatures,” the academy found.

In the 1980s, Dr. Thouless showed that the integers by which the conductivity of electricity could be measured were topological in their nature. Around that time, Dr. Haldane discovered how topological concepts could be used to understand the properties of chains of small magnets found in some materials.

“We now know of many topological phases, not only in thin layers and threads, but also in ordinary three-dimensional materials,” the academy said. “Over the last decade, this area has boosted front-line research in condensed matter physics, not least because of the hope that topological materials could be used in new generations of electronics and superconductors, or in future quantum computers.”

Michael S. Turner, a physicist at the University of Chicago, said by email that the work of the three prizewinners was “truly transformational, with long-term consequences both practical and fundamental.”

“It illustrates the importance and surprises associated with curiosity-driven research,” he added.

What Is Topology?

At a news conference in Stockholm, Thors Hans Hansson, a member of the Nobel physics committee, used a bagel, a pretzel and a cinnamon bun to explain topology. While the items vary across many variables, a topologist focuses only on the holes: The pretzel has two, the bagel has one, and the bun has none.

“Things like taste or shape or deformation can change continuously, but the number of holes — something that we call the topological invariant — can only change by integers, like 1, 2, 3, 0,” he said.

This topological insight turned out to be useful in understanding the conductance — the ease with which electric current flows through a substance — in certain two-dimensional materials at extremely low temperatures and in strong magnetic fields. While the research was largely theoretical, it could have practical applications for items like electronics, superconductors and even computers.

Who Are the Winners?

Dr. Thouless, 82, was born in Bearsden, Scotland, was an undergraduate at Cambridge University and received a Ph.D. in 1958 from Cornell. He taught mathematical physics at the University of Birmingham in England from 1965 to 1978, where he collaborated with Dr. Kosterlitz. He joined the University of Washington in Seattle in 1980, where he is now an emeritus professor.

Dr. Haldane, 65, was born in London. He received his Ph.D. from Cambridge, where he was also an undergraduate, in 1978. He worked at the Institut Laue-Langevin in Grenoble, France, the University of Southern California, Bell Laboratories and the University of California, San Diego, before joining Princeton in 1990.

Dr. Kosterlitz, 73, was born in Aberdeen, Scotland, and received his doctorate in high energy physics from Oxford University in 1969. He has worked at the University of Birmingham; at the Instituto di Fisica Teorica in Turin, Italy; and Cornell, Princeton, Bell Laboratories and Harvard.

Statements From the Winners

“I was very surprised and very gratified,” Dr. Haldane, whom the Nobel committee reached by phone Tuesday morning, told the news conference in Stockholm. “The work was a long time ago, but it’s only now that a lot of tremendous new discoveries are based on this original work, and have extended it.”

Dr. Kosterlitz told The Associated Press that he got the news while heading to lunch in Helsinki, where he is a visiting professor at Aalto University.

“I’m a little bit dazzled,” he said. “I’m still trying to take it in.”

He said he was in his 20s when he began studying two-dimensional materials and that his “complete ignorance” was an advantage in challenging the established science.

“I didn’t have any preconceived ideas,” he said. “I was young and stupid enough to take it on.”

Who Else Has Won a Nobel This Year?

Yoshinori Ohsumi, a Japanese cell biologist, was awarded the Nobel Prize in Physiology or Medicine on Monday for his discoveries on how cells recycle their content, a process known as autophagy, a Greek term for “self-eating.”

Who Won the 2015 Physics Nobel?

Takaaki Kajita and Arthur B. McDonald were named co-laureates last year for discovering that the enigmatic subatomic particles known as neutrinos have mass.