University of Miami
Department of Mathematics
College of Arts and Sciences

Lecture Series
Spring Semester 2019

Isolated Hypersurface Singularities of Analytic Varieties

presented by

Distinguished Professor Phillip Griffiths

Ungar Building, Room 506
4:00pm

Wednesday, April 3, 2019
Wednesday, April 10, 2019
Wednesday, April 17, 2019


Abstract

The study of isolated singularities of analytic varieties, especially in the case of isolated hypersurfaced singularities.

f(x0, .., xn) = 0

where f(x) is an analytic function, is a classical and very beautiful subject. It is one of significant current interest due to the role played by such singularities in moduli theory and birational geometry using the local to global principal. For example, one may study the extent to which the classical theory can be extended to the case where f is defined on a variety with cononical singularities. The theme of these lectures will be to show how elementary complex analysis in several variables can be used to prove and illustrate some of the main analyticand topological properties associated to an isolated singularity. This approach will be somewhat different from the traditional ones.

The plan of the lectures is

1. Local study of finate holomorphic mappings, essentially based on residues and local dualities

2. The Milnor fibration as a special case of the Lefschetz fibration; de Rham cohomology of an isolated singularity, analytic and topological interpretations of the Milnor number

3. Gauss-Manin connection and the monodromy theorem; determination of the spectrum of monodromy when f(x) is a weighted homogeneous polynominal

We will attempt to keep the lectures self-contained. Some basic familiarity with very elementary complex analysis, differential forms (the classical de Rham theorem) and elementary commutative algebra (the Koszul complex) will be helpful. Proofs will be sketched with the essential ideas isolated and references to the detailed and more general arguments given.


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.