



The Index Theorem
The
AtiyahSinger Index Theorem is considered one of the greatest achievements
of 20^{th} century mathematics. Unfortunately, the exact result is
impossible to state in layman's terms, but very roughly it provides a link
between differential equations and topology.
(See this paper for a professional introduction, and
this comment for a lighter treatment).
This result is the culmination of
a long chain of ideas, starting with Stokes Theorem, which
some readers may have encountered in undergraduate math courses, and passing
through Hodge theory and the HirzebruchRiemannRoch Theorem. Surprisingly, the
Index Theorem has numerous applications, not just in pure mathematics, but also
in theoretical physics. Indeed, this result, as well the tireless efforts of
Atiyah and Singer in general, have lead to a highly fruitful
crossfertilization between mathematics and theoretical physics, which has left
a profound impact on both disciplines.
The Abel Prize
The Abel Prize was created on the occasion the of the 200^{th} birthday
of Niels
Henrik Abel, a brilliant Norwegian mathematician who tragically died at a
young age just as his achievements were starting to receive due recognition.
The prize, worth NOK 6,000,000 (USD 875,000, EUR 710,000), is
awarded once per
year by the Norwegian Academy of Science and Letters to acknowledge outstanding
accomplishments in the field of mathematics. It was awarded for the first time last
year, when it went to
JeanPierre Serre.
As is well known, there is no Nobel Prize
in Mathematics. Instead, most people have looked to the Fields Medal as the
most prestigious award for mathematical achievement. The Fields Medal, however,
differs significantly from the Nobel Prize in that it is only awarded every
four years (at each meeting of the International Congress of Mathematicians),
to two, three or four recipients. Moreover, it is only awarded to
mathematicians not older than 40, in order to encourage further achievements.
This also has the pleasant side effect of rewarding recent work, rather than
work done half a century earlier, as is too often the case with Nobel
Prizes. Tellingly, both Atiyah (1966) and Serre (1954) have also won Fields
Medals.
The Abel Prize, by contrast, seems to resemble the Nobel Prize more closely. It
is awarded annually by a Scandinavian learned society (the Nobel Prize is
awarded by the Swedish Academy of Sciences), it is worth a considerable amount
of money, and so far all three recipients are over seventy years old. Happily,
this allows rewarding those (relatively few) mathematicians who missed the
Fields Medals due to their age. In particular, by 1966 Singer was already over
the age limit (he was born in 1924), and only Atiyah was awarded the Fields
medal for the AtiyahSinger Index Theorem.


