abel prize 2011,winner of abel prize 2011,Milnor wins 2011 Abel Prize

Tag : abel prize 2011,winner of abel prize 2011,Milnor wins 2011 Abel Prize

Milnor wins 2011 Abel Prize

Biography:

John Milnor was educated at Princeton University, receiving his A.B. in 1951. He began research at Princeton after graduating and, in 1953 before completing his doctoral studies, he was appointed to the faculty in Princeton.

In 1954 Milnor received his doctorate for his thesis Isotopy of Links written under Ralph Fox's supervision. Milnor remained on the staff at Princeton where he was an Alfred P Sloan fellow from 1955 until 1959. He was promoted to professor in 1960 then, in 1962, Milnor was appointed to the Henry Putman chair.

Milnor was awarded a Fields Medal at the 1962 International Congress of Mathematicians in Stockholm. His most remarkable achievement, which played a major role in the award of the Fields Medal, was his proof that a 7-dimensional sphere can have several differential structures. This work opened up the new field of differential topology.

Milnor showed that 28 different differentiable structures exist on the seven-dimensional sphere. He distinguished between these structures using numerical invariants based on the Todd polynomials. The Todd polynomials were first studied in algebraic geometry and it is surprising that they play this fundamental role in classification of manifolds. The reason that Milnor could use them to distinguish the differential properties of manifolds is because they have arithmetic properties, involving the Bernoulli numbers, which reflect in a deep and not fully understood way these differential properties.

The references [2] to [6] give a good indication of the wide influence of Milnor's work up to 1992 (when these articles were written). The article is a survey of Milnor's work in algebra, particularly in algebraic K-theory, where his work continues to have important influences.

The articlelooks at nine papers which Milnor had written on differential geometry. It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4π. Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group.

In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in [6]. He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. He also studied the Steenrod algebra and its dual, investigated the structure of Hopf algebras, and studied characteristic classes and their relation to mathematical physics.

Honours awarded to John Milnor

BMC plenary speaker 1958, 1978

Speaker at International Congress 1962

Fields Medal 1962

BMC morning speaker 1964

Hedrick lecturer 1965

AMS Colloquium Lecturer 1968

AMS Steele Prize 1982

Wolf Prize 1989

AMS Steele Prize 2004 

abel prize 2011