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Goro Nishida

A major international meeting on homotopy theory took place in Kinosaki, Japan, from July 28–August 1 2003, followed on August 4–8 by an intense satellite conference at the Nagoya Institute of Technology. This volume contains the Proceedings of those conferences. They, and this volume, are dedicated to Professor Gôrô Nishida on the occasion of his 60th birthday.

Nishida's earliest work grew out of the study of infinite loopspaces. His first paper (in 1968, on what came eventually to be known as the Nishida relations) accounts for interactions between Steenrod and Dyer–Lashof (Kudo–Araki) operations. This was followed by early work with H Toda on the extended power construction, which led in 1973 to his milestone proof of the nilpotence of positive-degree elements in the stable homotopy ring of spheres.

This result, whose echoes continue to reverberate today in work of Devinatz, Hopkins, Smith, and others on the chromatic picture, and in work on motives in algebraic geometry, stood at the time as an isolated beacon of hope in the (then very mysterious) world of stable homotopy theory. It, together with the Kahn-Priddy theorem, was one of the first signs that the subject possesses deep global properties – that it held structural secrets well beyond its already formidable computational aspects. Nishida next turned his attention to a circle of ideas surrounding the Segal conjecture, transfer homomorphisms, and stable splittings of classifying spaces of groups. The ideas in this series of papers have by now grown into a rich subfield of homotopy theory, with important contributions by Benson, Feshbach, Martino, Minami, Priddy, Webb, and many others; it continues today in (for example) the theory of p-compact groups. In recent years much of his work has been concerned with various aspects of elliptic cohomology. His deep insight from the early 90's, that work of Eichler and Shimura on modular forms, higher S1–transfers, and the diffeomorphism group of the two-torus are all intimately connected, is still not adequately understood; its exploitation may depend on new geometric ideas from the developing theory of elliptic objects.

Today in retrospect Nishida's thinking has always seemed very global and adventuresome – as he has been, personally, as well – reaching back at least as far as his early 70's postdoc in Manchester. His hospitality and kindness to visitors is legendary, and he has devoted enormous energies to opening the international community to younger researchers. This volume is dedicated to him by his many colleagues, in recognition of and admiration for his ideas and his work.

The NSF supported travel to this conference by more than twenty researchers from the US, under grant DMS 0080657; it was the greatest opportunity in a generation for younger algebraic topologists on both sides of the Pacific to confer about common interests. The conferences focused on the broad range of subjects which Professor Nishida has influenced, and which are still critical to research today: localization, periodicity in homotopy theory, higher K–theories, infinite loop spaces, homology and cohomology operations, modular forms, group cohomology, and stable homotopy of classifying spaces. The organizing committee on the Japanese side was headed by Professor Norihiko Minami of the Nagoya Institute of Technology (representing Professor Kazuhisa Shimakawa of Okayama University), assisted by Atsushi Yamaguchi of Osaka Prefecture University, Akihiro Oshita of the Osaka University of Economics, Dai Tamaki of Shinshu University, and Hirofumi Nakai of the Oshima College of Maritime Technology. Stewart Priddy of Northwestern, Jack Morava and W Stephen Wilson of Johns Hopkins, and Clarence Wilkerson of Purdue were the organizing committee on the US side.

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