Some of the world's top mathematicians and theoretical physicists gather on campus this weekend for a conference in honor of one of the pioneers of the field, Professor Albert Schwarz of the UC Davis mathematics department.
For 50 years, Schwarz has explored the extraordinary territory where theoretical physics and pure mathematics meet and blend.
"Physics is a very rich source of mathematical problems, and solutions to these problems are useful for physicists and for pure mathematics also," Schwarz said.
Schwarz has made a series of discoveries that solved physics problems while opening up new areas of mathematics, said Motohico Mulase, professor of mathematics and chair of the organizing committee.
For most of the 20th century, physicists developing a new theory would generally find that mathematicians had already created tools to study it, he said. In 1960, physicist Eugene Wigener called this the "unreasonable effectiveness of mathematics."
"The key contribution of Albert Schwarz is to reverse this relationship between mathematics and physics. He showed the world how useful physics is in geometry and topology as a tool of discovering new ideas and results," Mulase said.
Speakers at the conference, which started Thursday and runs through Sunday, include mathematicians and theoretical physicists and five winners of the Fields Medal, considered the Nobel Prize of mathematics. They are Alain Connes, College de France, Paris; Vaughan Jones, UC Berkeley; Maxim Kontsevich, Institut des Hautes Etudes Scientifique, France; Sergey Novikov, University of Maryland; and Edward Witten, Institute of Advanced Study, Princeton.
"It's true testimony to the international reputation of Albert Schwarz that such prominent speakers, stars in the field of mathematical physics, would come to this conference," said Winston Ko, dean of the division of mathematical and physical sciences.
Born in 1934, Schwarz spent his research career in the Soviet Union until he and his family left in 1989. His parents were persecuted during Stalin's purges in 1937; his mother was exiled to Kazakhstan, where his grandmother took Schwarz in 1941. In 1948, they were allowed to leave exile and settled in Ivanovo, about 200 miles from Moscow.
Unable to attend Moscow University because of his parents' arrest, Schwarz entered the Ivanovo Pedagogical Institute in 1951. He graduated in 1955, after Stalin's death, and was able to attend graduate school at Moscow University, gaining the equivalent of a Ph.D. in 1958.
Blocked by the KGB from accepting job offers at two scientific institutes, Schwarz began his working career at Voronezh University, where he stayed until 1964 when he joined the Department of Theoretical Physics at the Moscow Physical Engineering Institute.
"In 1964, it was quite unusual for a mathematician to work in a physics department," he said. "Interaction between physicists and mathematicians was almost non-existent -- now it's commonplace."
Before Mikhail Gorbachev's "Perestroika" reforms, Schwarz was able to travel abroad only twice, to Bulgaria and Czechoslovakia. In 1988 he visited Poland, and in 1989 accepted an invitation to the International Center for Theoretical Physics in Trieste, Italy. He returned to Russia briefly and in July 1989, he and his family left the Soviet Union for good.
"I knew I would try and stay in the West, I didn't know if I would succeed," Schwarz said.
Schwarz spent time at the Institute of Advanced Study at Princeton, Harvard University and MIT before joining the mathematics department at UC Davis in 1990.
Schwarz's work has covered a range of topics at the intersection of mathematics and theoretical physics. His contributions have mostly been based on applying mathematical tools such as topology, noncommutative geometry and homological algebra to quantum field theory, string theory and M-theory -- areas which, he said with some understatement, are difficult to explain.
His recent work has been on string theory or M-theory. Once described as "a piece of 21st century physics that fell by chance into the 20th century," string theory holds that elementary particles are made up of loops of vibrating "string." Fundamental particles such as electrons, photons, quarks and neutrinos arise from different vibrations of the same type of string.
At one time, physicists hoped that string theory could be the unifying "theory of everything" tying together forces from the subatomic to the cosmic. It's since become clear that string theory needs to be modified, leading to M-theory, Schwarz said.
More recently, the idea that strings are fundamental objects of the universe has disappeared, and researchers think that there may be other fundamental objects or that strings are themselves made of something else.
"There are many questions to ask and many problems to solve," Schwarz said. "I believe that M-theory will be included in another theory and after several iterations, we will get to the fundamental equations."
It could also be the case that there is a much simpler way to explain the fundamentals of the universe and that M-theory represents a detour, he said. In that case, M-theory will either disappear or become part of a much larger structure of theory.