John Willard Milnor American mathematician. Date of Birth: 20.02.1931 Country: USA

1. Biography of John Willard Milnor
2. The Fary-Milnor Theorem
3. Exotic Spheres

Biography of John Willard Milnor

John Willard Milnor is an American mathematician who made significant contributions to differential geometry, topology, and algebra. Born on March 23, 1931 in the small town of Orange, New Jersey, Milnor displayed a talent for mathematics from an early age. He studied at Princeton University and began his serious scientific work under the guidance of his mentor, Ralph Fox.

In 1954, Milnor defended his dissertation on "Isotopy of Knots" and remained at his alma mater to continue his research. In 1962, he was awarded the Fields Medal, the most prestigious award in mathematics, for his contributions to differential topology. In 1988, he moved to Stony Brook University in New York, where he continues to work to this day.

Despite his unremarkable personal biography, Milnor has had a profound impact on modern mathematics throughout his more than 60-year career. His books, such as "Morse Theory" written when he was just 24 years old, are included in the recommended reading list for young Russian scientists preparing for their exams in the field of geometry and topology.

The Fary-Milnor Theorem

One of Milnor's notable results is the Fary-Milnor theorem, which belongs to the field of knot theory. Knots in mathematics are similar to regular knots, with the difference that the ends of a knot are always considered to be connected. In 1950, Milnor discovered that if a knot is not too twisted, meaning its total curvature is small enough, then it can be untied. The Fary-Milnor theorem states that if the total curvature of a knot is less than or equal to 4π, the knot is trivial and can be untangled.

Exotic Spheres

Milnor's second significant discovery, which contributed to his receiving the Fields Medal, is the existence of exotic spheres. In 1956, he found a seven-dimensional manifold that resembled a sphere but was not diffeomorphic to it. This discovery challenged the prevailing belief that differentiable objects were more refined than topological ones. Milnor showed that there exist objects that are topologically equivalent but not diffeomorphic. These exotic spheres have since become a fundamental concept in differential topology.

In addition to these achievements, Milnor's contributions include the Milnor hypothesis, the Milnor fibration, and many other objects named in recognition of their exceptional importance. The Norwegian Academy of Science made the right decision in awarding Milnor the Abel Prize, as his groundbreaking work has solidified his status as a living legend, a great mathematician, and a contemporary of ours.