Special Years

2008-2009: Analytic and Algebraic Geometry: Multiplier Ideals    

Coordinators: Herbert Clemens and Jeffery McNeal

A particular technique from PDE's was brought to bear on problems in algebraic geometry, with remarkable success.

The basic version of this technique consists of establishing certain crucial inequalities, that do not universally hold, by studying the set of functions or operators which "tame'' the inequality, i.e. the set of multipliers for a particular inequality. In many situations, it turns out that the set of multipliers possesses a remarkable number of unexpected properties, for example they form an ideal. These properties then allow one to formulate conditions, of both algebraic and geometric character, which give situations where the original inequality holds.

In the hands of algebraic geometers, this basic idea has been expanded upon and greatly abstracted. And it has been successfully applied to many questions of current interest, including the invariance of plurigenera, the construction of Kähler-Einstein manifolds on algebraic manifolds, and Fujita's conjecture, among others. These successes, however, point to even more possible uses of the multiplier ideal notion. By bringing together the analysts and algebraic geometers who work in directions close to multiplier ideals -- but who speak about the concept in different languages and who rarely interact with one another -- we expect that the range of application for this general idea will be greatly broadened and enrich both the algebraic and analytic sides in the process.  More math details

Mechanics of the program.

The program is designed to follow up on the Park City Mathematics Institute (PCMI) which took place in the summer of 2008 on the same mathematical topic, and it will incorporate extended interaction with the FRG in Algebraic Geometry at the University of Michigan.

We first outline the research and graduate education aspects of the Park City program of 2008. The mathematicians listed below gave a week-long course for graduate students (and their colleagues) at PCMI2008:

Bo Berndtsson (Chalmers University of Technology): Introduction to -methods in complex geometry.

John D'Angelo (University of Illinois at Urbana-Champaign): Real and complex varieties and orders of contact.

Jean-Pierre Demailly (Université de Grenoble): TBA

Christopher Hacon (University of Utah): Finite generation of the canonical ring.

János Kollár (Princeton University): TBA

Robert Lazarsfeld (University of Michigan): Introduction to multiplier ideals.

Mircea Mustata (University of Michigan): Resolution of singularities.

Dror Varolin (SUNY at Stony Brook): Invariance of plurigenera. Skoda's theorems.

Each PCMI course was accompanied by a problem session for less advanced graduate students and research working groups (RWG) for more advanced ones. During the three weeks of PCMI2008, senior mathematicians (possibly including the ones above) also gave seminar-style lectures on topics connected to the courses above which are on the research frontier.

Six one-week workshops are taking place at OSU during the 2008-09 academic year.  Graduate student participants in PCMI 2008, as well as postdocs and faculty PCMI participants. are cordially invited to participate in these week-long workshops, which will take place in the Mathematics Department on the Columbus campus of Ohio State.  Limited funding is available.

The following workshops have taken place or are already scheduled:

October 6-10, 2008:  "Minimal model program," led by Professor János Kollár of Princeton University
November 17-21:  "L_2 methods and the del-bar equation," led by Professor Bo Berndtsson, Chalmers University of Technology, Sweden
January 12-16, 2009:  "L_2 methods in complex geometry," led by Prof. Dror Varolin, S.U.N.Y., Stony Brook
February 23-27, 2009:  "Multiplier ideals and their applications," led by Professors Robert Lazarsfeld and Mircea Mustata, University of Michigan, Ann Arbor
March 16-20, 2009:  "Higher-dimensional minimal model program," led by Professor Christopher Hacon, University of Utah

Format for each week will be very informal.  Students will be organized into small groups to work on problems from the summer course.  One constant will be a daily 2-hour morning session in which the content of the lecturer's PCMI2008 GSS course is reviewed and grad students report on their work on the problems.  Lecturers will also give one or more seminars or colloquium talks during their visit.

At least one more workshop will be scheduled for the Spring Quarter.  Details to follow once plans are final.