**Edward Witten** was born into a Jewish family. His mother was Lorraine W Witten while his father, Louis Witten, was a theoretical physicist specializing in gravitation and general relativity. His brother, Matt Witten, has achieved fame as a screenwriter and television producer.

Witten studied at Brandeis University and received his B.A. in 1971. From there he went to Princeton receiving his M.A. in 1974 and his Ph.D. in 1976.

After completing his doctorate, Witten went to Harvard where he was postdoctoral fellow during session 1976-77 and then a Junior Fellow from 1977 to 1980. In September 1980 Witten was appointed professor of Physics at Princeton. He was awarded a MacArthur Fellowship in 1982 and remained as professor of Physics at Princeton until 1987 when he was appointed as a Professor in the School of Natural Sciences at the Institute for Advanced Study.

Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes in [3]:-

Although he is definitely a physicist(as his list of publications clearly shows)his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.

Speaking at the American Mathematical Society Centennial Symposium in 1988, Witten explained the relation between geometry and theoretical physics:-

It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics. ... Of course, quantum physics had from the beginning a marked influence in many areas of mathematics - functional analysis and representation theory, to mention just two. ... Several important influences have brought about a change in this situation. One of the principal influences was the recognition - clearly established by the middle1970s - of the central role of nonabelian gauge theory in elementary particle physics. The other main influence came from the emerging study of supersymmetry and string theory.

In his study of these areas of theoretical physics, Witten has achieved a level of mathematics which has led him to be awarded the highest honour that a mathematician can receive, namely a Fields Medal. He received the medal at the International Congress of Mathematicians which was held in Kyoto, Japan in 1990. The Proceedings of the Congress contains two articles describing Witten's mathematical work which led to the award. The main tribute is the article [3] by Atiyah, but Atiyah could not be in Kyoto to deliver the address so the address at the Congress was delivered by Faddeev [5] who quotes freely from Atiyah [3].

The first major contribution which led to Witten's Fields Medal was his simpler proof of the positive mass conjecture which had led to a Fields Medal for Yau in 1982. Gawedzki and Soulé describe this work by Witten, which appeared in 1981, in [9]:-

The proof ... employed in a subtle way the idea of supersymmetry. This became the centrepiece of many of Witten's subsequent works...

One of Witten's subsequent works was a paper which Atiyah singles out for special mention in [3], namely *Supersymmetry and Morse theory* which appeared in the *Journal of differential geometry* in 1984. Atiyah writes that this paper is:-

... obligatory reading for geometers interested in understanding modern quantum field theory. It also contains a brilliant proof of the classic Morse inequalities, relating critical points to homology. ... Witten explains that "supersymmetric quantum mechanics" is just Hodge-de Rham theory. The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the Hodge-de Rham theory of infinite dimensional manifolds. It is a measure of Witten's mastery of the field that he has been able to make intelligent and skilful use of this difficult point of view in much of his subsequent work.

Since this highly influential paper, the ideas in it have become of central importance in the study of differential geometry. Further new ideas of fundamental importance were introduced by Witten and described in [9]:-

Witten subsequently gave a string interpretation of the elliptic genus and provided arguments for its rigidity ... Another piece of new mathematics stemmed from Witten's papers on global gravitational anomalies. ... In recent years, Witten focused his attention on topological quantum field theories. These correspond to Lagrangians ... formally giving manifold invariants. Witten described these in terms of the invariants of Donaldson and Floer(extending the earlier ideas of Atiyah)and generalised the Jones knot polynomial...

The authors of [9] sum up Witten's contributions to mathematics:-

Although mostly not in the form of completed proofs, Witten's ideas have triggered major mathematical developments by the force of their vision and their conceptual clarity, his main discoveries soon becoming theorems. His Fields Medal at the1990International Congress of Mathematicians acknowledged the growing impact of his work on contemporary mathematics.

Atiyah, in [3], expresses the same ideas in the following way:-

... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming.

Witten has received an amazingly long list of awards, in addition to the Fields Medal, which shows the exceptionally high regard in which his contributions are held. He has been awarded honorary degrees from Brandeis University (1988), the Hebrew University of Jerusalem (1993), Columbia University, New York (1996), the University of Southern California (2004), Johns Hopkins University (2005), and Harvard University (2005). He has received the Einstein Medal from the Einstein Society of Berne, Switzerland (1985), the Award for Physical and Mathematical Sciences from the New York Academy of Sciences (1985), the Dirac Medal from the International Center for Theoretical Physics (1985), the Alan T Waterman Award from the National Science Foundation (1986), the Madison Medal from Princeton University (1992), the New Jersey Pride Award (1996), the Award of the Golden Plate from the American Academy of Achievement (1997), the Klein Medal from Stockholm University (1998), the Dannie Heineman Prize from the American Institute of Physics (1998), the Nemmers Prize in Mathematics from Northwestern University (2000), the Clay Research Award from the Clay Mathematics Institute (2001), the Shalom Award from Americans for Peace Now (2002), the National Medal of Science (2003), the Pythagoras Award from Crotone, Italy (2005), the Harvey Prize from the Technion, Israel (2006), the Poincaré Prize from the International Association Of Mathematical Physics (2006), and the Crafoord Prize in Mathematics from The Royal Swedish Academy of Sciences (2008). He has been elected a Fellow of numerous academies and societies such as American Academy of Arts and Sciences (1984), the American Physical Society (1984), the National Academy of Sciences (1988), the American Philosophical Society (1994), the Royal Society of London (1998), the Academy of Sciences of Paris (2000), and the Pontifical Academy (2006).

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

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