Record

Reference numberEC/1994/42
LevelItem
TitleHirzebruch, Friedrich Ernst Peter: certificate of election to the Royal Society
Date1993
DescriptionCertificate of Candidate for Election to Foreign Membership. Citation typed on separate piece of paper, then pasted onto certificate
CitationHirzebruch is the leading German mathematician of his generation, and more than anyone else he has been the architect of the post-war rebuilding of mathematics in Germany. He has also played an important role in organising mathematical cooperation internationally, and most recently he has been very active in connection with the academic problems of Eastern Europe.
In his earliest works he was a pioneer in the introduction of topological methods into algebraic geometry, but in the process he changed the direction of both fields His first famous theorem related the signature of a manifold to its characteristic cohomology classes. On one side this was the beginning of a development which led into functional analysis on manifolds, and culminated in the Atiyah-Singer index theorem. On the other side it provided the crucial clue for the topological classification of manifolds and the discovery of exotic differential structures. Simultaneously with the signature theorem Hirzebruch proved a general version of the Riemann Roch theorem which has been a fundamental tool in all subsequent algebraic geometry.
Hirzebruch's study of characteristic cohomology clsses lead him to study the topology of Lie groups in general, an area which he elucidated once and for all in three long papers (joint with A. Borel)
A little later, together with Atiyah, he invented topological K-theory, the first example of an "exotic" cohomology theory, a concept which has been central in algebraic topology ever since (its great success came at once when Adams used it to settle the classical question of the number of independent vector fields on a sphere.)
After the remarkable decade in which all the preceding work was done Hirzebruch turned to a rather different area, which occupied him for the next fifteen years. This was the study of Hilbert modular surfaces, which are fascinating structures both for algebraic geometry and for number thoery. His work on modular surfaces was very concrete and detailed, and he eventually succeeded in analysing almost every aspect of them: the geometry of the configurations of curves on them, the explicit resolution of their singularities, the algebras of modular forms they give rise to. They turned out to provide crucial examples for mathematicians with many different interests.
In the last ten years Hirzebruch's work has turned in a somewhat new direction with his increasing interest in so-called "elliptic genera". This has its origins in his much earlier wortk on the multiplicativity of the signature in fibre bundles, and on the vanishing of the A-genus for manifolds with a circle action, and it is also related to his interest in modular forms. What had made the area reawaken, however is the recent discovery, mainly die to Witten, that some of the more exotic genera which Hirzebruch had introduced in his earliest work actually arise naturally in quantum field theory.
Access statusClosed
Fellows associated with this archive
CodeNameDates
NA4724Hirzebruch; Friedrich Ernst Peter (1927 - 2012)1927 - 2012
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