The cause was lung cancer, his family said.

Although he began his career studying engineering, Dr. Bott, who was born in Hungary, turned to mathematics as a graduate student in the 1940's and later defined a mathematician as "someone who likes to get at the root of things." He taught at Harvard from 1959 until 1999.

Beginning in the 1960's, in a long and fruitful collaboration with Sir Michael Atiyah of the University of Edinburgh, Dr. Bott worked on refinements of mathematical index theory to find topological ways of investigating solutions to differential equations. All are aspects in theoretical mathematics.

The collaboration yielded the Atiyah-Bott fixed-point theorem, which in part shows that a mathematical map has a fixed point and also provides a means to count the number of fixed points on a given map.

Dr. Bott was also widely known for earlier work, when he developed what became known as the Bott periodicity theorem in 1959, the importance of which some mathematicians have compared to the discovery of the periodic table of the elements.

"It came as a great surprise and provided a whole set of new tools," Sir Michael said. "Astonishingly, it has also proved useful in the fundamental physics of string theory."

Loring W. Tu, an associate professor of mathematics at Tufts University in Medford, Mass., said the theory "describes the topology, or shapes, of rotations in spaces of large dimension and explains that the rotations are not as complicated as once thought."

Dr. Tu, who also collaborated with Dr. Bott and wrote about his research, added, "The theory helped shed light on aspects of geometry and was a big discovery."

In other work, Dr. Bott contributed to the Borel-Weil-Bott theorem and studied foliations, which are a type of differential equation. More recently, his research was directed toward developing mathematical tools to aid physicists working to reconcile theories of general and quantum relativity.

Robert D. MacPherson, a professor of mathematics at the Institute for Advanced Study in Princeton, N.J., and a former student of Dr. Bott's, said his research "represents an attempt to present mathematics as a unified thing and to bring ideas together - itself a remarkable achievement."

Raoul Harry Bott was born in Budapest. He received undergraduate and master's degrees from McGill University in Montreal, and his doctorate from the Carnegie Institute of Technology, part of Carnegie Mellon, in 1949.

He joined Harvard after earlier appointments at the Institute for Advanced Study and the University of Michigan. He became an American citizen in the 1950's.

In 1990, the American Mathematical Society awarded Dr. Bott its Leroy P. Steele Prize for Lifetime Achievement. He received the National Medal of Science in 1987 and won the Wolf Foundation Prize in Mathematics in 2000. He was a member of the National Academy of Sciences.

Dr. Bott is survived by his wife of 58 years, Phyllis. The couple lived in Cambridge, Mass., before recently moving to Carlsbad.

He is also survived by a son, Anthony, of Harwich, Mass.; three daughters, Candace Bott of Cambridge, Jocelyn Scott of Rancho Santa Fe, Calif., and Renee Bott of Berkeley, Calif.; and nine grandchildren.

By JEREMY PEARCE, January 8, 2006 © The New York Times Company