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The authors use methods from birational geometry to study the Hodge
filtration on the localization along a hypersurface. This
filtration leads to a sequence of ideal sheaves, called Hodge
ideals, the first of which is a multiplier ideal. They analyze
their local and global properties, and use them for applications
related to the singularities and Hodge theory of hypersurfaces and
their complements.
This is Part 1 of a two-volume set. Since Oscar Zariski organized a
meeting in 1954, there has been a major algebraic geometry meeting
every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985),
Santa Cruz (1995), and Seattle (2005). The American Mathematical
Society has supported these summer institutes for over 50 years.
Their proceedings volumes have been extremely influential,
summarizing the state of algebraic geometry at the time and
pointing to future developments. The most recent Summer Institute
in Algebraic Geometry was held July 2015 at the University of Utah
in Salt Lake City, sponsored by the AMS with the collaboration of
the Clay Mathematics Institute. This volume includes surveys
growing out of plenary lectures and seminar talks during the
meeting. Some present a broad overview of their topics, while
others develop a distinctive perspective on an emerging topic.
Topics span both complex algebraic geometry and arithmetic
questions, specifically, analytic techniques, enumerative geometry,
moduli theory, derived categories, birational geometry, tropical
geometry, Diophantine questions, geometric representation theory,
characteristic $p$ and $p$-adic tools, etc. The resulting articles
will be important references in these areas for years to come.
This is Part 2 of a two-volume set. Since Oscar Zariski organized a
meeting in 1954, there has been a major algebraic geometry meeting
every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985),
Santa Cruz (1995), and Seattle (2005). The American Mathematical
Society has supported these summer institutes for over 50 years.
Their proceedings volumes have been extremely influential,
summarizing the state of algebraic geometry at the time and
pointing to future developments. The most recent Summer Institute
in Algebraic Geometry was held July 2015 at the University of Utah
in Salt Lake City, sponsored by the AMS with the collaboration of
the Clay Mathematics Institute. This volume includes surveys
growing out of plenary lectures and seminar talks during the
meeting. Some present a broad overview of their topics, while
others develop a distinctive perspective on an emerging topic.
Topics span both complex algebraic geometry and arithmetic
questions, specifically, analytic techniques, enumerative geometry,
moduli theory, derived categories, birational geometry, tropical
geometry, Diophantine questions, geometric representation theory,
characteristic $p$ and $p$-adic tools, etc. The resulting articles
will be important references in these areas for years to come.
Contemporary research in algebraic geometry is the focus of this
collection, which presents articles on modern aspects of the
subject. The list of topics covered is a roll-call of some of the
most important and active themes in this thriving area of
mathematics: the reader will find articles on birational geometry,
vanishing theorems, complex geometry and Hodge theory, free
resolutions and syzygies, derived categories, invariant theory,
moduli spaces, and related topics, all written by leading experts.
The articles, which have an expository flavour, present an overall
picture of current research in algebraic geometry, making this book
essential for researchers and graduate students. This volume is the
outcome of the conference Recent Advances in Algebraic Geometry,
held in Ann Arbor, Michigan, to honour Rob Lazarsfeld's many
contributions to the subject on the occasion of his 60th birthday.
Algebraic geometry is one of the most diverse fields of research in
mathematics. It has had an incredible evolution over the past
century, with new subfields constantly branching off and
spectacular progress in certain directions, and at the same time,
with many fundamental unsolved problems still to be tackled. In the
spring of 2009 the first main workshop of the MSRI algebraic
geometry program served as an introductory panorama of current
progress in the field, addressed to both beginners and experts.
This volume reflects that spirit, offering expository overviews of
the state of the art in many areas of algebraic geometry.
Prerequisites are kept to a minimum, making the book accessible to
a broad range of mathematicians. Many chapters present approaches
to long-standing open problems by means of modern techniques
currently under development and contain questions and conjectures
to help spur future research.
Algebraic geometry is one of the most diverse fields of research in
mathematics. It has had an incredible evolution over the past
century, with new subfields constantly branching off and
spectacular progress in certain directions, and at the same time,
with many fundamental unsolved problems still to be tackled. In the
spring of 2009 the first main workshop of the MSRI algebraic
geometry program served as an introductory panorama of current
progress in the field, addressed to both beginners and experts.
This volume reflects that spirit, offering expository overviews of
the state of the art in many areas of algebraic geometry.
Prerequisites are kept to a minimum, making the book accessible to
a broad range of mathematicians. Many chapters present approaches
to long-standing open problems by means of modern techniques
currently under development and contain questions and conjectures
to help spur future research.
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