Lecture Series by Phillip Griffiths - Fall 2023

Abstract: Atypical Hodge Loci
Recent work by a number of people has shown that for a general family of smooth varieties of dimension at least 3 the non-trivial Hodge loci have strictly bigger dimension than expected. This talk will sketch the proof of their result and explain why it is true.

Talk based on the paper [BKU] and related works given in the references in that work, and on extensive discussions with Mark Green and Colleen Robles.

Abstract: Period Mappings for Anti-canonical Pairs
Anti-canonical pairs (Y, D) are logarithmic K3 surfaces. It is well known that they have a rich geometry. A recent result, whose proof was motivated by mirror-symmetry, establishes a conjecture by Looijenga giving conditions for smoothability of the cusp obtained by contracting D. A central ingredient in the proof is a global Torelli theorem using the mixed Hodge structure on H2(Y − D). In this talk we will formulate and sketch the proof of this result.

Based on the works of Looijenga [L81], Friedman[F16], Engel-Friedman [EF21], and Gross-Hacking-Keel [GHK15]. Presentation including notations, largely follows [F16].

Abstract: A Smoothness Result in Deformation Theory
Deformation theory is a fundamental part of algebraic geometry. It’s goal is to construct the local moduli space of algebro-geometric objects (varieties, bundles, maps between varieties etc.). The 1st or-der deformations are typically represented by a cohomology group. The obstructions to extending a 1st order deformation to higher order are in another cohomology group, one which is generally differicult to compute. In this talk the basic result expressing in terms of ordinary topological cohomology groups for Kähler manifolds sufficient conditions for the obstructions to vanish will be formulated and proved.

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.