Integral Closure, Multiplier Ideals and Cores
December 17 to December 21, 2006
at the
American Institute of Mathematics,
Palo Alto, California
organized by
Alberto Corso,
Claudia Polini,
and Bernd Ulrich
Original Announcement
This workshop will be devoted to questions related to the notion
of integral closure of ideals.
The generalization to ideals of the basic concepts of integral extensions
and integral closures of rings can be traced back to the
fundamental work of Zariski and Rees in local algebra. Loosely speaking,
the integral closure of an ideal I is an ideal
contained in the radical of I that shares a number of finer
properties with I. Determining the integral closure of I is a difficult task,
which essentially amounts
to finding solutions in the ring itself of special polynomial
equations whose coefficients belong to higher and higher powers
of I.
The aspects intimately connected to the integral closure
that we are planning to focus on are:
computation of the integral closure and its complexity;
multiplicities and equisingularity theory;
cores of ideals and BrianconSkoda type theorems;
multiplier ideals and test ideals;
multiplier ideals and jet schemes.
More concretely, some of the specific questions/open problems that we will address
during the workshop are:

find effective methods to compute (parts of) the integral closure of
an ideal (or, more generally, of a submodule of a free module);

find tests to detect when an ideal (or, more generally, a submodule of
a free module) is integrally closed;

find a relationship between the core and the adjoint for
integrally closed and monomial ideals;

determine whether the adjoint has the subadditivity property;

find necessary and/or sufficient conditions for integrally closed ideals
in regular local rings of dimension at least 3 to be multiplier ideals;

further explore the relationship between test ideals and multiplier
ideals.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
Reports of the working groups
Lecture notes
Continuous closure and variants of integral closure, by Hochster
Open problems on powers of ideals, by Huneke
Equisingularity and integral closure, by Kleiman
Local syzygies of multiplier ideals, by Lee
A vanishing theorem for finitely supported ideals in regular local rings, by Lipman
FPure thresholds and log canonical thresholds, by Tagaki
Squarefree monomial ideals and hypergraphs, by Trung
Rees criteria, by Ulrich
On Fthresholds (ring theoretic aspects), by Watanabe
Papers arising from the workshop: