Volume 1. Works of Philip A. Griffiths with commentary: Differential geometry and Hodge Theory (1983-2014) -- volume 2. Selected Works of Philip A. Griffiths with commentary: Algebraic cycles (2003-2007)

Containing four parts such as Analytic Geometry, Algebraic Geometry, Variations of Hodge Structures, and Differential Systems that are organized according to the subject matter, this title provides the reader with a panoramic view of important and exciting mathematics during the second half of the 20th century.

The development of algebraic topology in the 1950's and 1960's was deeply influenced by the work of Milnor. In this collection of papers the reader finds those original papers and some previously unpublished works. The book is divided into four parts: Homotopy Theory, Homology and Cohomology, Manifolds, and Expository Papers. Introductions to each part provide some historical context and subsequent development. Of particular interest are the articles on classifying spaces, the Steenrod algebra, the introductory notes on foliations and the surveys of work on the Poincare conjecture. Together with the previously published volumes I-III of the Collected Works by John Milnor, volume IV provides a rich portion of the most important developments in geometry and topology from those decades. This volume is highly recommended to a broad mathematical audience, and, in particular, to young mathematicians who will certainly benefit from their acquaintance with Milnor's mode of thinking and writing.

An introduction to Griffiths' theory of period maps and domains, focused on algebraic, group-theoretic and differential geometric aspects.

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck’s algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne’s theorem on absolute Hodge cycles), and variation of mixed Hodge structures. The contributors include Patrick Brosnan, James Carlson, Eduardo Cattani, François Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark L. Green, Phillip A. Griffiths, Matt Kerr, Lê Dũng Tráng, Luca Migliorini, Jacob P. Murre, Christian Schnell, and Loring W. Tu.

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.