**University of Miami**

Department of Mathematics

*College of Arts and Sciences*

Department of Mathematics

*College of Arts and Sciences*

**Lecture SeriesSpring Semester 2018**

**What is Complex Algebraic Geometry?**

**presented by**

**Distinguished Professor Phillip Griffiths**

Ungar Building, Room 506

5:00 pm

Thursday, March 8^{th}, 2018

Thursday, March 22^{nd}, 2018

Thursday, March 29^{th}, 2018

Wednesday, April 11^{th}, 2018 (REVISED)

**Outline**

- Origins; algebraic functions and their integrals (modern algebraic geometry began with a result in calculus)
- Analytic methods; PDEs and differential geometry (most of the deepest results about complex algebraic varieties such as Kodiara vanishing and Hard Lefschetz require analysis for their proofs)
- Topology and Hodge Theory (the basic invariant of a complex algebraic variety is the Hodge structure on its cohomology; from this flows the extraordinary properties of the topology of algebraic varieties)
- What is the Hodge conjecture and why hasn't it been proved? (the Hodge conjecture has an arithmetic aspect that is not yet understood)

**Abstract**

Algebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field k When k = C the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal and illustrative account of some answers to the question in the title. Every attempt will be made to have the talks accessible to an audience of graduate students and post docs.

*Some Information:*

*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.