不动点理论及应用

Copyright © 2006 Philip R. Heath et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
The International Conference on Nielsen Theory and Related Topics took place from
June 28 through July 2, 2004 at Memorial University, St. John’s, Newfoundland, Canada.
This was the 13th such conference in a series that began in 1977 with a conference in
Oberwolfach, Germany.
Nielsen theory is named after Jakob Nielsen who, in the 1920s, turned the focus of
fixed point theory fromthe existence of fixed points (as, e.g., in the famous Lefschetz fixed
point theorem) to the problem of estimating the actual number of such points within the
homotopy class of a given map. He did this by introducing what is now called the Nielsen
number of a self map, a homotopy invariant lower bound for the number of fixed points
of the map. After important initial contributions by Reidemeister andWecken, there was
little activity in Nielsen theory until the 1960s when a breakthrough by Boju Jiang allowed
for easy calculations of the Nielsen number for maps on Lie groups and some other interesting
kinds of spaces. It was these and other important examples that would guide
the direction of research. As the present collection of papers illustrates, the frontiers of
the subject now involve an impressive variety and interplay of algebraic and geometric
techniques, on a wide class of spaces. Consequently, there continue to be an increasing
number of interesting areas for future investigations, both in the areas of computation
and the development of new invariants.
As research in Nielsen theory progressed, its concern with fixed points expanded, on
the one hand, into related issues such as coincidences, periodic points, and roots and, on
the other, into various refinements for restricted classes of maps and homotopies such as
those that occur in the fiber space, relative and equivariant contexts. Just about all of these
aspects of Nielsen theory were represented by the talks in Newfoundland. The conferencewas attended by 36 people from14 different countries and five of the six inhabitable continents;
this comprises almost all mathematicians who are currently active in research in
Nielsen theory. The participants displayed a range of experience ranging from graduate
students to people who had attended the previous Canadian Nielsen theory conference,
in Sherbrooke, Quebec, in 1980. The group also included a number of experienced mathematical
researchers who are new to Nielsen theory. Their varied expertise is providing
important new directions for research in our subject.
The major financial support for the conference was furnished by Canada’s Atlantic Association
for Research in theMathematical Sciences (AARMS) which, in turn, is partially
funded by Memorial University of Newfoundland (MUN). We are profoundly grateful
for the generous support from AARMS. Substantial funding for the attendance of participants
from developing countries came from the Commission on Development and
Exchanges (CDE) of the International Mathematical Union (IMU). We thank Prof. Herbert
Clemens, Secretary of the CDE, and also the Executive Committee of the IMU which
modified its rules in order to support these mathematicians at a conference not held in
a developing country. Special thanks are due to the secretaries of the MUN Mathematics
Department, in particular to Wanda Heath and Ros English, for their outstanding help
in organizing the conference.
We are also grateful to Prof. Ravi Agarwal and the publishers of the journal “Fixed
Point Theory and Applications” for the opportunity to celebrate the occasion of the Newfoundland
conference by publishing the collection of research papers in Nielsen theory
that occupy this issue.