In its current incarnation, a regulator is a map from the algebraic K-theory
of an algebraic variety to a suitable cohomology theory such as ´ etale cohomology
or Deligne cohomology. Regulators come in many different forms, according to
the context. For instance, the Borel regulator is the higher-degree analogue of the
Dirichlet regulator, considered as a map on algebraic K-theory in degree one. On
the other hand, in Riemann surface theory, the regulators might involve Abelian
integrals and Jacobians, extending the ideas of the 19th century analytic number
theorists and geometers. Over the past 20 years, there has been renewed interest
in regulators, stimulated by the trail-blazing works of S. Bloch, A. Beilinson and
the more recent revolutionary work of V. Voevodsky. The contributions in this
proceedings provide an important snap-shot of the current developments in the
field by leading researchers.
In May of 1998, there was an Oberwolfach meeting entitled “Regulators” (or-
ganised by S. Bloch, M. Kolster, P. Schneider and V. Snaith), which involved a
number of participants working on regulators from different perspectives. The
Oberwolfach workshop was regarded as a great success, and from it sprang a series
of conferences under the same name. Regulators II (organized by J. D. Lewis and
V. Snaith) was subsequently held in December of 2005 at the Banff International
Research Station. The atmosphere at that second meeting could best be described
as “electric”. Indeed there was a fruitful exchange of ideas that led to numerous
interesting developments in the subject.
It was decided that Regulators III would be held again on European soil, in
July of 2010 in the city of Barcelona. Each day would center around four high level
talks pushing the boundaries of this subject area, with ample time for discussions
and collaborations. The end of an intense work day would be tempered by the
spectacular backdrop of this scenic city. The feeling one had was one of enthusiasm
and inspiration.
Over a glass of wine with colleagues, the discussion turned to where and when
to hold the Regulators IV conference. But then our minds would wander back to
the present, with a feeling of accomplishment. Indeed this is a conference one is
not likely to ever forget!
A generous thanks to our financial sponsors: the ANR (France), the NSF
(USA), the CSIC (Spain), the Spanish “Ministerio de Ciencia e Innovaci´ on” through
grants “I-MATH” (Consolider Ingenio) and MTM2009-064050-E, the AGAUR of
the “Generalitat de Catalunya” through grant 2009 ARCS2 00080, and the
“Facultat de Matem` atiques de la Universitat de Barcelona” through direct sup-
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