| eBook ISBN: | 978-1-4704-3072-6 |
| Product Code: | FIC/38.E |
| List Price: | $136.00 |
| MAA Member Price: | $122.40 |
| AMS Member Price: | $108.80 |
| eBook ISBN: | 978-1-4704-3072-6 |
| Product Code: | FIC/38.E |
| List Price: | $136.00 |
| MAA Member Price: | $122.40 |
| AMS Member Price: | $108.80 |
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Book DetailsFields Institute CommunicationsVolume: 38; 2003; 367 ppMSC: Primary 14
The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others.
The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others.
The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds.
The book is suitable for researchers interested in mirror symmetry and string theory.
Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipGraduate students and research mathematicians interested in mirror symmetry and string theory.
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Table of Contents
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Geometric methods
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Victor Batyrev and Evgeny Materov — Mixed toric residues and Calabi-Yau complete intersections
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Li Chiang and Shi-shyr Roan — Crepant resolutions of $\mathbb {C}^n/A_1(n)$ and flops of $n$-folders for $n=4,5$
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Pedro del Angel and Stefan Müller-Stach — Picard-Fuchs equations, integrable systems and higher algebraic K-theory
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Shinobu Hosono — Counting BPS states via holomorphic anomaly equations
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James Lewis — Regulators of Chow cycles on Calabi-Yau varieties
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Arithmetic methods
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Philip Candelas, Xenia de la Ossa and Fernando Rodriguez-Villegas — Calabi-Yau manifolds over finite fields, II
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Luis Dieulefait and Jayanta Manoharmayum — Modularity of rigid Calabi-Yau threefolds over $\mathbb {Q}$
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Yasuhiro Goto — $K3$ surfaces with symplectic group actions
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Tetsushi Ito — Birational smooth minimal models have equal Hodge numbers in all dimensions
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Bong Lian and Shing-Tung Yau — The $n$th root of the mirror map
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Ling Long — On a Shioda-Inose structure of a family of K3 surfaces
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Monika Lynker, Vipul Periwal and Rolf Schimmrigk — Black hole attractor varieties and complex multiplication
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Fernando Rodriguez-Villegas — Hypergeometric families of Calabi-Yau manifolds
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Rolf Schimmrigk — Aspects of conformal field theory from Calabi-Yau arithmetic
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Jan Stienstra — Ordinary Calabi-Yau-3 crystals
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Jan Stienstra — The ordinary limit for varieties over $\mathbb {Z}[x_1,\ldots ,x_r]$
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Noriko Yui — Update on the modularity of Calabi-Yau varieties with appendix by Helena Verrill
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Noriko Yui and James Lewis — Problems
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The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others.
The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others.
The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds.
The book is suitable for researchers interested in mirror symmetry and string theory.
Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Graduate students and research mathematicians interested in mirror symmetry and string theory.
-
Geometric methods
-
Victor Batyrev and Evgeny Materov — Mixed toric residues and Calabi-Yau complete intersections
-
Li Chiang and Shi-shyr Roan — Crepant resolutions of $\mathbb {C}^n/A_1(n)$ and flops of $n$-folders for $n=4,5$
-
Pedro del Angel and Stefan Müller-Stach — Picard-Fuchs equations, integrable systems and higher algebraic K-theory
-
Shinobu Hosono — Counting BPS states via holomorphic anomaly equations
-
James Lewis — Regulators of Chow cycles on Calabi-Yau varieties
-
Arithmetic methods
-
Philip Candelas, Xenia de la Ossa and Fernando Rodriguez-Villegas — Calabi-Yau manifolds over finite fields, II
-
Luis Dieulefait and Jayanta Manoharmayum — Modularity of rigid Calabi-Yau threefolds over $\mathbb {Q}$
-
Yasuhiro Goto — $K3$ surfaces with symplectic group actions
-
Tetsushi Ito — Birational smooth minimal models have equal Hodge numbers in all dimensions
-
Bong Lian and Shing-Tung Yau — The $n$th root of the mirror map
-
Ling Long — On a Shioda-Inose structure of a family of K3 surfaces
-
Monika Lynker, Vipul Periwal and Rolf Schimmrigk — Black hole attractor varieties and complex multiplication
-
Fernando Rodriguez-Villegas — Hypergeometric families of Calabi-Yau manifolds
-
Rolf Schimmrigk — Aspects of conformal field theory from Calabi-Yau arithmetic
-
Jan Stienstra — Ordinary Calabi-Yau-3 crystals
-
Jan Stienstra — The ordinary limit for varieties over $\mathbb {Z}[x_1,\ldots ,x_r]$
-
Noriko Yui — Update on the modularity of Calabi-Yau varieties with appendix by Helena Verrill
-
Noriko Yui and James Lewis — Problems
