John Milnor: Spheres

Winner of the 2011 Abel Prize for mathematics John Milnor presented an historical account of work on topological and differential spheres in a special colloquium.
Mathematicians have analyzed the possible shapes and topologies of space in many dimensions since the late 1800s when the field of topology originated. Milnor's lecture traced the work of leading topologists analyzing and dissecting the possible shapes, or topologies of space, in various dimensions. Spheres have been a central theme of topology for the last 60 years and Milnor initiated that study. He said that the first real breakthrough in high-dimensional cases came in 1961 and that topologists had success in dimensions two and below and five and above, leaving third- and fourth-dimensional shapes, which proved most difficult, to be solved. He described the work he did with French mathematician Michel Kervaire in the 1960s on 7-D (dimensional) spheres that a number of other contributors carried on when he and Kervaire stopped collaborating. Although mathematicians continued to work on the classification of spheres in dimensions higher than four, the 4-D case is still open, said Milnor.