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This volume contains articles related to the work of the Simons
Collaboration "Arithmetic Geometry, Number Theory, and
Computation." The papers present mathematical results and
algorithms necessary for the development of large-scale databases
like the L-functions and Modular Forms Database (LMFDB). The
authors aim to develop systematic tools for analyzing Diophantine
properties of curves, surfaces, and abelian varieties over number
fields and finite fields. The articles also explore examples
important for future research. Specific topics include algebraic
varieties over finite fields the Chabauty-Coleman method modular
forms rational points on curves of small genus S-unit equations and
integral points.
The contributions in this book explore various contexts in which
the derived category of coherent sheaves on a variety determines
some of its arithmetic. This setting provides new geometric tools
for interpreting elements of the Brauer group. With a view towards
future arithmetic applications, the book extends a number of
powerful tools for analyzing rational points on elliptic curves,
e.g., isogenies among curves, torsion points, modular curves, and
the resulting descent techniques, as well as higher-dimensional
varieties like K3 surfaces. Inspired by the rapid recent advances
in our understanding of K3 surfaces, the book is intended to foster
cross-pollination between the fields of complex algebraic geometry
and number theory. Contributors: * Nicolas Addington * Benjamin
Antieau * Kenneth Ascher * Asher Auel * Fedor Bogomolov *
Jean-Louis Colliot-Thelene * Krishna Dasaratha * Brendan Hassett *
Colin Ingalls * Marti Lahoz * Emanuele Macri * Kelly McKinnie *
Andrew Obus * Ekin Ozman * Raman Parimala * Alexander Perry * Alena
Pirutka * Justin Sawon * Alexei N. Skorobogatov * Paolo Stellari *
Sho Tanimoto * Hugh Thomas * Yuri Tschinkel * Anthony
Varilly-Alvarado * Bianca Viray * Rong Zhou
Based on the Simons Symposia held in 2015, the proceedings in this
volume focus on rational curves on higher-dimensional algebraic
varieties and applications of the theory of curves to arithmetic
problems. There has been significant progress in this field with
major new results, which have given new impetus to the study of
rational curves and spaces of rational curves on K3 surfaces and
their higher-dimensional generalizations. One main recent insight
the book covers is the idea that the geometry of rational curves is
tightly coupled to properties of derived categories of sheaves on
K3 surfaces. The implementation of this idea led to proofs of
long-standing conjectures concerning birational properties of
holomorphic symplectic varieties, which in turn should yield new
theorems in arithmetic. This proceedings volume covers these new
insights in detail.
The contributions in this book explore various contexts in which
the derived category of coherent sheaves on a variety determines
some of its arithmetic. This setting provides new geometric tools
for interpreting elements of the Brauer group. With a view towards
future arithmetic applications, the book extends a number of
powerful tools for analyzing rational points on elliptic curves,
e.g., isogenies among curves, torsion points, modular curves, and
the resulting descent techniques, as well as higher-dimensional
varieties like K3 surfaces. Inspired by the rapid recent advances
in our understanding of K3 surfaces, the book is intended to foster
cross-pollination between the fields of complex algebraic geometry
and number theory. Contributors: * Nicolas Addington * Benjamin
Antieau * Kenneth Ascher * Asher Auel * Fedor Bogomolov *
Jean-Louis Colliot-Thelene * Krishna Dasaratha * Brendan Hassett *
Colin Ingalls * Marti Lahoz * Emanuele Macri * Kelly McKinnie *
Andrew Obus * Ekin Ozman * Raman Parimala * Alexander Perry * Alena
Pirutka * Justin Sawon * Alexei N. Skorobogatov * Paolo Stellari *
Sho Tanimoto * Hugh Thomas * Yuri Tschinkel * Anthony
Varilly-Alvarado * Bianca Viray * Rong Zhou
Based on the Simons Symposia held in 2015, the proceedings in this
volume focus on rational curves on higher-dimensional algebraic
varieties and applications of the theory of curves to arithmetic
problems. There has been significant progress in this field with
major new results, which have given new impetus to the study of
rational curves and spaces of rational curves on K3 surfaces and
their higher-dimensional generalizations. One main recent insight
the book covers is the idea that the geometry of rational curves is
tightly coupled to properties of derived categories of sheaves on
K3 surfaces. The implementation of this idea led to proofs of
long-standing conjectures concerning birational properties of
holomorphic symplectic varieties, which in turn should yield new
theorems in arithmetic. This proceedings volume covers these new
insights in detail.
This volume contains articles related to the work of the Simons
Collaboration “Arithmetic Geometry, Number Theory, and
Computation.” The papers present mathematical results and
algorithms necessary for the development of large-scale databases
like the L-functions and Modular Forms Database (LMFDB). The
authors aim to develop systematic tools for analyzing
Diophantine properties of curves, surfaces, and abelian varieties
over number fields and finite fields. The articles also explore
examples important for future research. Specific topics include●
algebraic varieties over finite fields● the Chabauty-Coleman
method● modular forms● rational points on curves of small
genus● S-unit equations and integral points.
Providing an overview of the state of the art on rationality
questions in algebraic geometry, this volume gives an update on the
most recent developments. It offers a comprehensive introduction to
this fascinating topic, and will certainly become an essential
reference for anybody working in the field. Rationality problems
are of fundamental importance both in algebra and algebraic
geometry. Historically, rationality problems motivated significant
developments in the theory of abelian integrals, Riemann surfaces
and the Abel-Jacobi map, among other areas, and they have strong
links with modern notions such as moduli spaces, Hodge theory,
algebraic cycles and derived categories. This text is aimed at
researchers and graduate students in algebraic geometry.
This volume resulted from the conference A Celebration of Algebraic
Geometry, which was held at Harvard University from August 25-28,
2011, in honour of Joe Harris' 60th birthday. Harris is famous
around the world for his lively textbooks and enthusiastic
teaching, as well as for his seminal research contributions. The
articles are written in this spirit: clear, original, engaging,
enlivened by examples, and accessible to young mathematicians. The
articles in this volume focus on the moduli space of curves and
more general varieties, commutative algebra, invariant theory,
enumerative geometry both classical and modern, rationally
connected and Fano varieties, Hodge theory and abelian varieties,
and Calabi-Yau and hyperkahler manifolds. Taken together, they
present a comprehensive view of the long frontier of current
knowledge in algebraic geometry.
Algebraic geometry, central to pure mathematics, has important
applications in such fields as engineering, computer science,
statistics and computational biology, which exploit the
computational algorithms that the theory provides. Users get the
full benefit, however, when they know something of the underlying
theory, as well as basic procedures and facts. This book is a
systematic introduction to the central concepts of algebraic
geometry most useful for computation. Written for advanced
undergraduate and graduate students in mathematics and researchers
in application areas, it focuses on specific examples and restricts
development of formalism to what is needed to address these
examples. In particular, it introduces the notion of Grobner bases
early on and develops algorithms for almost everything covered. It
is based on courses given over the past five years in a large
interdisciplinary programme in computational algebraic geometry at
Rice University, spanning mathematics, computer science,
biomathematics and bioinformatics.
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