In his prizewinning work, Dr. Thouless studied materials so thin that they could be considered two-dimensional. To these, he applied quantum physics and topology, a branch of mathematics.

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Topology places a premium on recognizing objects with gross similarities in physical structure, objects that retain their distinctive aspects despite all sorts of bends and twists and folds.

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In the world of daily reality, substances may exist in liquid, solid and vapor phases. In the two-dimensional, low-temperature realms that Dr. Thouless studied, new and unusual phases of matter were thought to exist.

Great interest was aroused by one in particular: the phase characterized by the extraordinary ability to conduct electricity. This was known as superconductivity.

At the time Dr. Thouless came upon the scene, it seemed to science that transitions in phase in thin films could not be adequately explained. It was thought that random fluctuations, such as those occurring in low-temperature systems, would prevent order among the atoms and molecules and thereby preclude such phenomena as superconductivity.

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To the study of thin materials at low temperatures, and to their ability or inability to conduct electricity, Dr. Thouless applied the methods of topology and thereby produced, in the words of the Nobel organization, “surprising results” and significant developments.

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Dr. Thouless, then affiliated with the University of Washington in Seattle, shared the 2016 Nobel with two other physicists from the United Kingdom who also worked in the United States: F. Duncan Haldane of Princeton University and J. Michael Kosterlitz of Brown University.

Dr. Thouless and Kosterlitz were credited with discovering a transition between phases of thin materials that was based on viewing these materials as systems of spinning vortices. At low temperatures, the vortices would pair up, but as the temperature rose, they would split off and enter a new phase.

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Beyond the sense of astonishment conveyed by these results, they raised the curtain on new areas of study and showed the way to important new ideas in physics, the Nobel committee noted. The theories devised by Dr. Thouless and Kosterlitz brought about an entirely new understanding of phase transitions and of new phases of matter fabricated into thin films, the committee said.

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David James Thouless was born in Bearsden, Scotland, on Sept. 21, 1934, and grew up in Cambridge. In a way, his career in theoretical physics, a demanding mental discipline, might have seemed foreordained.

His father, Robert, held a PhD in psychology and wrote “Straight and Crooked Thinking,” a book about the rigorous analysis of flawed arguments. In the United States, it was assigned for years in college classes on rhetoric and argument: The American title was “How to Think Straight.”

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His mother, the former Priscilla Gorton, held degrees in literature and taught English at the college level. Interesting guests often showed up at the family home. Among them was the celebrated philosopher Ludwig Wittgenstein.

According to his Nobel biography, Dr. Thouless taught himself reading and writing with the aid of a housekeeper, and when he did his army service, on long marches he and a friend entertained themselves with games of mental chess.

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At Cambridge, he received a bachelor’s degree in 1955 and met physicist and future Nobel laureate Hans Bethe, who was on a sabbatical from Cornell University. Bethe invited the young graduate to come to the United States to study under him.

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At Cornell, Dr. Thouless recalled, a course in experimental physics was required.

“I do not think any of my experiments came out right,” he once said. “But apparently the explanations I gave of what had gone wrong and what I needed to do about it were sufficiently convincing that I got the highest grade in the course.”

He said a second semester was not required of him.

Physics legend has it that Dr. Thouless asked Bethe for something to work on for his PhD research and then returned to Bethe two years later, finished thesis in hand.

It was not quite that way. But Bethe was a busy man, and Dr. Thouless recalled that talking with Bethe as infrequently as once a month left him with three months’ worth of ideas. He received his doctorate in theoretical physics in 1958.

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At Cornell, he married Margaret Scrase, a biology student. They had three children, Michael, Christopher and Helen.

Dr. Thouless worked at the Lawrence Berkeley National Laboratory in California on a postdoctoral fellowship. He was a professor at the University of Birmingham in England from 1965 to 1978 and joined the University of Washington faculty in 1980. He became a professor emeritus in 2003. According to Trinity Hall, he and his wife returned to live in Cambridge in 2014.

In addition to the Nobel, Dr. Thouless’s major honors in physics included the Wolf Prize in 1990, the Paul Dirac Medal in 1993 and the Lars Onsager Prize in 2000.