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.
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978-0-8218-3355-1 (9780821833551)
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Schweitzer Klassifikation
Geometric methods: Mixed toric residues and Calabi-Yau complete intersections by V. V. Batyrev and E. N. Materov Crepant resolutions of $\mathbb{C}^n/A_1(n)$ and flops of $n$-folders for $n=4,5$ by L. Chiang and S.-s. Roan Picard-Fuchs equations, integrable systems and higher algebraic K-theory by P. L. del Angel and S. Muller-Stach Counting BPS states via holomorphic anomaly equations by S. Hosono Regulators of Chow cycles on Calabi-Yau varieties by J. D. Lewis Arithmetic methods: Calabi-Yau manifolds over finite fields, II by P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas Modularity of rigid Calabi-Yau threefolds over $\mathbb{Q}$ by L. Dieulefait and J. Manoharmayum $K3$ surfaces with symplectic group actions by Y. Goto Birational smooth minimal models have equal Hodge numbers in all dimensions by T. Ito The $n$th root of the mirror map by B. H. Lian and S.-T. Yau On a Shioda-Inose structure of a family of K3 surfaces by L. Long Black hole attractor varieties and complex multiplication by M. Lynker, V. Periwal, and R. Schimmrigk Hypergeometric families of Calabi-Yau manifolds by F. Rodriguez-Villegas Aspects of conformal field theory from Calabi-Yau arithmetic by R. Schimmrigk Ordinary Calabi-Yau-3 crystals by J. Stienstra The ordinary limit for varieties over $\mathbb{Z}[x_1,\ldots,x_r]$ by J. Stienstra Update on the modularity of Calabi-Yau varieties with appendix by Helena Verrill by N. Yui Problems by N. Yui and J. D. Lewis.
