This grant is to support US participants traveling to the Summer School on Hodge Theory that will take place at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, from June 14 to July 3, 2010. The objectives of the Summer School are to give an overview of modern Hodge Theory with an emphasis on integrating the geometric and algebraic aspects and on highlighting the increasing interest on the arithmetic/geometric aspects of the theory. The hope is to present to students the overall framework of the subject and to introduce them to promising areas for future study and research. Thus this summer school on Hodge Theory will provide a unique opportunity for graduate students and young researchers to enter one of the most active areas of current research in algebraic geometry.

Hodge theory is an important area of differential and algebraic geometry with deep connections to many fields of mathematics and physics. Hodge theory has involved techniques drawn from algebraic, arithmetic, analytic and diff erential geometry, as well as from topology and partial di fferential equations. This summer school on Hodge Theory will provide a unique opportunity for graduate students and young researchers to enter one of the most active areas of current research in algebraic geometry.

Project Report

This grant supported the participation of graduate students and young researchers from the US in the Summer School Conference on Hodge Theory and Related Topics which took place at The Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, from June 4 to July 2, 2010. It brought together 106 participants from 33 different countries and 23 School and Conference lecturers for a three-week introduction to Hodge Theory, an area of Algebraic Geometry with applications to Theoretical Physics, Number Theory and Representation Theory. The School provided a unique opportunity for graduate students and young researchers to enter one of the most active areas of current research in algebraic geometry. The variety of techniques required for the study of Hodge Theory greatly contributes to the richness of the subject but has had the unfortunate effect that its study is not part of the typical graduate school curriculum and learning the basics has become a very formidable task. The ICTP Summer School went a long way toward introducing a new generation of students and young researchers to Hodge Theory. The School lecturers were all leading researchers in the field including Phillip Griffiths (IAS), James Carlson (Clay Mathematics Institute), Mark Green (UCLA), Jacob Murre (Leiden), Mark de Cataldo (SUNY, Stony Brook), Luca Migliorini (Bologna), Fouad El Zein (Paris), Loring Tu (Tufts), Eduardo Cattani (UMass Amherst), Patrick Brosnan (University of British Columbia) as well as a younger researchres such as Matt Kerr (Washington University), Christian Schnell (University of Illinois Chicago), and François Charles (ENS, Paris). After a week of basic and introductory courses on Kähler manifolds, cohomology of algebraic varieties, mixed Hodge structures, variations of Hodge structure, and Hodge maps, the participants were introduced to three main areas of current research: the study of normal functions, their zero locus and singularities; the theory of Mumford-Tate groups and domains with its beautiful interactions with geometry, number theory and representation theory; absolute Hodge cycles and the Beilinson-Bloch conjecture. Much of this work was presented at the School for the first time; this is the case for the work of M. Green, P. Griffiths, and M. Kerr on Mumford-Tate Groups and Shimura Varieties and the recent work of C. Voisin on Absolute Hodge Cyles presented in the lectures by C. Schnell, F. Charles, and M. Kerr. All the School lecturers prepared Lecture Notes which were made available to the participants and which will be edited and published as part of a volume currently in preparation. The activity concluded with a three-day conference on current developments and applications.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1001125
Program Officer
Tie Luo
Project Start
Project End
Budget Start
2010-04-01
Budget End
2011-03-31
Support Year
Fiscal Year
2010
Total Cost
$24,997
Indirect Cost
Name
University of Massachusetts Amherst
Department
Type
DUNS #
City
Amherst
State
MA
Country
United States
Zip Code
01003