Michael Hartley Freedman

Born: 21 April 1951 in Los Angeles, California, USA

Michael Freedman's parents Benedict and Nancy Freedman are both quite famous. His mother, Nancy Mars, was born in Chicago in 1920 and had some acting roles before attending the Chicago Art Institute, Los Angeles City College, and the University of Southern California. She married Benedict Freedman on 29 June 1941. Benedict Freedman's father, David, was born in Romania. Benedict was a talented mathematician, musician and writer. He studied at Curtiss-Wright Technical Institute at Glendale, California, graduating with a degree in aeronautical engineering. He taught aeronautical engineering at Curtiss-Wright during the 1940s but had a parallel career as a scriptwriter of radio shows, drama critic and newspaper editor. Benedict and Nancy Freedman are joint authors of several well-known novels. They had three children, Johanna, Michael and Deborah.
Michael showed exceptional talents in mathematics as he grew up. However, he also enjoyed painting in an expressionist style and when he entered the University of California at Berkeley in 1968 he still had not decided between mathematics and art. Quite quickly he made a firm decision to study mathematics but, near the end of his first year of study, he decided to apply to do graduate studies at Princeton University. Freedman enjoyed playing Go and he knew that the mathematician Ralph Fox at Princeton was a champion Go player. He read Fox's 1963 text Introduction to Knot Theory and included conjectures of his own in his application to Princeton in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.

After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.

Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.

Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaré conjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remained open until settled by G Perelman who was offered the 2006 Fields medal for his proof.

Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-

Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.
Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-
After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.

Besides these results about closed simply connected four manifolds, Freedman also proved:

  1. Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 × R.

  2. There is a nonsmoothable closed four manifold.

  3. The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.

Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.

In his reply Freedman thanked his teachers (whom he said included his students) and also gave some fascinating views on mathematics [3]:-
My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.
In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-
In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.
In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.

Freedman continued to hold the Charles Lee Powell Professorship of Mathematics at the University of California at San Diego until 1998 when he left the academic world to take up an appointment with Station Q, a Microsoft research group working on topological quantum computing. In doing so, Freedman became the first Fields Medallist to leave the academic world to work for a company. Freedman became the director of Station Q and to see a little of the topics that he worked on we will look at a few of the titles of the papers he has written over the succeeding ten years. First we note that he gave an invited address to the International Congress of Mathematicians in Berlin in 1998 on Topological views on computational complexity. Then he published papers such as: Quantum computation and the localization of modular functors (2001); Projective plane and planar quantum codes (2001); Poly-locality in quantum computing (2002); Simulation of topological field theories by quantum computers (2002); Topological quantum computation (2003); Approximate Counting and Quantum Computation (2005); Topological quantum computation (2006); Topological quantum computing with only one mobile quasiparticle (2006); Interacting anyons in topological quantum liquids: The golden chain (2008); Measurement-Only Topological Quantum Computation (2008); and Topological Phase in a Quantum Gravity Model (2008). He also published some papers which were not related to quantum computing such as Extension of incompressible surfaces on the boundaries of 3-manifolds (2000), Diameters of Homogeneous Spaces (2003), and Covering a nontaming knot by the unlink (2007).

Article by: J J O'Connor and E F Robertson

September 2009
MacTutor History of Mathematics