Lecture Series by Phillip Griffiths - Fall 2023

Abstract: Hodge structures were introduced in the 1930's by Hodge as an extension and generalization of the Hodge structure on the cohomology of non-singular algebraic curves and surfaces that were introduced in the 19th and early 20th centuries by Riemann, Picard, Poincaré and Lefschetz. In particular one of Hodge's objectives was to give a proof of the "hard Lefschetz" theorem, established by Picard for algebraic surfaces but for which the Lefschetz argument for the general case was incomplete. With the introduction by Deligne of mixed Hodge structures for the cohomology of general algebraic varieties one now has for them a basic and subtle invariant.

The study of moduli of algebraic varieties and of related objects such as singularities, vector bundles, and representations of the fundamental group has emerged in the second half of the 20th century continuing up through the present time as a major topic in algebraic geometry. It is natural to use Hodge theory as a tool for the study of moduli, and that is the subject of this these talks.

More specifically, a major interest centers around the study of completions of a moduli space M. What is a natural class of objects to add to the generally smooth ones parametrized by M? It is well known that singular limits of smooth varieties are frequently easier to analyze than the smooth ones. Degenerating a variety may simplify it; the issue is to find a natural class of limits that can be worked with.

It is natural to utilize the results and methods of Hodge theory in the study of moduli. Here the important point is that one has an extensive and deep understanding of how Hodge structures degenerate; the goal of these talks is to describe some of how this understanding may be brought to bear on moduli questions.

Previous and current lecture notes can be found at www.math.miami.edu/~pg

Some Information:

Phillip Griffiths
Member of the National Academy of Sciences

Dr. Phillip Griffiths is a College of Arts and Sciences Distinguished Scholar in Mathematics. He received his B.S. from Wake Forest University in 1959 and his Ph.D. from Princeton University in 1962. He served as the Institute for Advanced Study as Director from 1991until 2003, as Professor of Mathematics from 2004 until 2009, and as Professor Emeritus since 2009. He has served as the Chair of its Science Initiative Group since 1999. He was Provost and James B. Duke Professor of Mathematics at Duke University from 1983 to 1991. He has also served on the faculties of the University of California at Berkeley, Princeton University and Harvard University.

Dr. Griffiths is one of the world's foremost experts in algebraic geometry and was inducted into the National Academy of Science in 1979 and the American Academy of Arts and Sciences in 1995. Among his many honors, Dr. Griffiths is the recipient of the Chern Medal from the International Mathematical Union (2014), the Steele Prize for Lifetime Achievement from the American Mathematical Society (2014), the Brouwer Prize from the Royal Dutch Mathematical Society (2008) and the Wolf Foundation Prize in Mathematics (2008). He was a Guggenheim Foundation Fellow from 1980 until 1982.

Dr. Griffiths has served on many important advisory boards and committees throughout his career including the Board of Trustees for the Mathematical Sciences Research Institute (2008-2013; Chair 2010-2013), the Board of Directors of Banker’s Trust New York (1994-1999), the Board of Directors of Oppenheimer Funds (1999-2013), the Carnegie-IAS Commission on Mathematics and Science Education (Chair 2007-2009), and the Scientific Committee of the Beijing International Center for Mathematical Research (2010-2013). From 2002 to 2005 he was the Distinguished Presidential Fellow for International Affairs for the US National Academy of Sciences and from 2001 to 2010 Senior Advisor to the Andrew W. Mellon Foundation.