University of Miami
Department of Mathematics
College of Arts and Sciences
Spring Semester 2017
The Role of Curvature in Algebraic Geometry:
Some History and Recent Issues
Distinguished Professor Phillip Griffiths
Ungar Building, Room 402
Wednesday, February 22nd, 2017
Wednesday, March 1st, 2017
Wednesday, March 8th, 2017
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set.
There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures "The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping" and "The moduli space for varieties of general type is itself of log-general type". The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular.
In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
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.