University of Miami
Department of Mathematics
College of Arts and Sciences

Lecture Series
Spring Semester 2021

Topics in the Geometric Applications of Hodge Theory

presented by

Distinguished Professor Phillip Griffiths

Zoom Meeting ID: 944 7631 3405
https://miami.zoom.us/j/94476313405?pwd=YnROQkZrbHF0Q0hOTmdHYlVBNmpQdz09
Passcode: 933425

3:30pm - 4:30pm

Friday, April 9, 2021
Wednesday, April 14, 2021
Wednesday, April 21, 2021


Abstract

  1. Mixed Hodge structures
        a. Definition (linear algebra)
        b. Extension data (more linear algebra)
        c. Differential constraint on the extension data
        d. Examples
  2. Limiting mixed Hodge structures
        a. Definition (more linear algebra)
        b. Geometric structures on the extension data
        c. Computational algorithm for the geometric case
        d. Further examples
  3. Applications
        a. Definition of the 1st variation
        b. Examples – isolated singularities with finite monodromy
        c. Examples – elliptic singularities

The classification of algebraic varieties is a central part of algebraic geometry. Among the common tools that are employed are birational geometry, especially the study of singularities, and Hodge theory. The latter is frequently used as a sort of "black box": the cohomology of an algebraic variety has a functorial mixed Hodge structure and the formal properties of the corresponding category have very strong consequences.

On the other hand, geometric structures arise naturally in Hodge theory and one aspect of this will be the focus of these lectures. This aspect arises from the fact that mixed Hodge structures have extension data constructed by linear algebra. However limiting missed Hodge structures that arise when singular varieties are smoothed are also constructed by linear algebra but from this construction there is an associated geometric structure, and it is this property and its uses, illustrated by examples, that these talks will be centered around. The lectures will be elementary, mostly consisting of linear algebra constructions and we will assume no knowledge of Hodge theory or algebraic geometry, and the examples will be mostly given by pictures.

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.