eBook ISBN:  9781470430726 
Product Code:  FIC/38.E 
List Price:  $136.00 
MAA Member Price:  $122.40 
AMS Member Price:  $108.80 
eBook ISBN:  9781470430726 
Product Code:  FIC/38.E 
List Price:  $136.00 
MAA Member Price:  $122.40 
AMS Member Price:  $108.80 

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 infinitedimensional Lie algebras among others.
The developments in physics stimulated the interest of mathematicians in CalabiYau 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 CalabiYau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for CalabiYau threefolds defined over the rationals, the BlochBeilinson conjectures, regulator maps of higher algebraic cycles, PicardFuchs 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 zetafunctions and Lseries of mirror pairs of CalabiYau threefolds.
The book is suitable for researchers interested in mirror symmetry and string theory.
Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipGraduate students and research mathematicians interested in mirror symmetry and string theory.

Table of Contents

Geometric methods

Victor Batyrev and Evgeny Materov — Mixed toric residues and CalabiYau complete intersections

Li Chiang and Shishyr Roan — Crepant resolutions of $\mathbb {C}^n/A_1(n)$ and flops of $n$folders for $n=4,5$

Pedro del Angel and Stefan MüllerStach — PicardFuchs equations, integrable systems and higher algebraic Ktheory

Shinobu Hosono — Counting BPS states via holomorphic anomaly equations

James Lewis — Regulators of Chow cycles on CalabiYau varieties

Arithmetic methods

Philip Candelas, Xenia de la Ossa and Fernando RodriguezVillegas — CalabiYau manifolds over finite fields, II

Luis Dieulefait and Jayanta Manoharmayum — Modularity of rigid CalabiYau 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 ShingTung Yau — The $n$th root of the mirror map

Ling Long — On a ShiodaInose structure of a family of K3 surfaces

Monika Lynker, Vipul Periwal and Rolf Schimmrigk — Black hole attractor varieties and complex multiplication

Fernando RodriguezVillegas — Hypergeometric families of CalabiYau manifolds

Rolf Schimmrigk — Aspects of conformal field theory from CalabiYau arithmetic

Jan Stienstra — Ordinary CalabiYau3 crystals

Jan Stienstra — The ordinary limit for varieties over $\mathbb {Z}[x_1,\ldots ,x_r]$

Noriko Yui — Update on the modularity of CalabiYau varieties with appendix by Helena Verrill

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 infinitedimensional Lie algebras among others.
The developments in physics stimulated the interest of mathematicians in CalabiYau 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 CalabiYau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for CalabiYau threefolds defined over the rationals, the BlochBeilinson conjectures, regulator maps of higher algebraic cycles, PicardFuchs 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 zetafunctions and Lseries of mirror pairs of CalabiYau threefolds.
The book is suitable for researchers interested in mirror symmetry and string theory.
Titles in this series are copublished 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 CalabiYau complete intersections

Li Chiang and Shishyr Roan — Crepant resolutions of $\mathbb {C}^n/A_1(n)$ and flops of $n$folders for $n=4,5$

Pedro del Angel and Stefan MüllerStach — PicardFuchs equations, integrable systems and higher algebraic Ktheory

Shinobu Hosono — Counting BPS states via holomorphic anomaly equations

James Lewis — Regulators of Chow cycles on CalabiYau varieties

Arithmetic methods

Philip Candelas, Xenia de la Ossa and Fernando RodriguezVillegas — CalabiYau manifolds over finite fields, II

Luis Dieulefait and Jayanta Manoharmayum — Modularity of rigid CalabiYau 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 ShingTung Yau — The $n$th root of the mirror map

Ling Long — On a ShiodaInose structure of a family of K3 surfaces

Monika Lynker, Vipul Periwal and Rolf Schimmrigk — Black hole attractor varieties and complex multiplication

Fernando RodriguezVillegas — Hypergeometric families of CalabiYau manifolds

Rolf Schimmrigk — Aspects of conformal field theory from CalabiYau arithmetic

Jan Stienstra — Ordinary CalabiYau3 crystals

Jan Stienstra — The ordinary limit for varieties over $\mathbb {Z}[x_1,\ldots ,x_r]$

Noriko Yui — Update on the modularity of CalabiYau varieties with appendix by Helena Verrill

Noriko Yui and James Lewis — Problems