Integral Closure, Multiplier Ideals and Cores

This workshop, sponsored by AIM and the NSF, 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 Briancon-Skoda 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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