Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations. Based on the conference, Institute Mittag-Leffler, Stockholm, Sweden, July 13–18, 2015.

*(English)*Zbl 1398.14003
Advanced Lectures in Mathematics (ALM) 42. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-363-0/pbk). ii, 619 p. (2018).

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This volume consists of expository papers on the four topics in its title, written by experts from around the world, and is the first to put forth a comprehensive discussion of these topics, and of the relations between them. As such, it is valuable as an introduction for beginners, and as a reference for mathematicians in general.

The articles of this volume will be reviewed individually.

Indexed articles:

Beukers, Frits, Hypergeometric functions, from Riemann till present, 1-19 [Zbl 1405.33002]

Doran, Brent; Doran, Charles F.; Harder, Andrew, Picard-Fuchs uniformization of modular subvarieties, 21-54 [Zbl 1409.32018]

Doran, Connemara, Poincaré’s path to uniformization, 55-79 [Zbl 1409.30037]

van der Geer, Gerard, Exploring modular forms and the cohomology of local systems on moduli spaces by counting points, 81-109 [Zbl 1446.11122]

Ji, Lizhen, Moduli spaces of compact Riemann surfaces: their complex structures and an overview of major results, 111-156 [Zbl 1409.32007]

Lazić, Vladimir; Oguiso, Keiji; Peternell, Thomas, The Morrison-Kawamata cone conjecture and abundance on Ricci flat manifolds, 157-185 [Zbl 1409.32010]

Lin, Chang-Shou, Conformal geometry and the Painlevé VI equation, 187-217 [Zbl 1410.34265]

Lu, Wenxuan, Calabi-Yau metrics: mirror symmetry Hitchin systems and instanton corrections, 219-235 [Zbl 1409.32016]

Papadopoulos, Athanase, Quasiconformal mappings, from Ptolemy’s Geography to the work of Teichmüller, 237-314 [Zbl 1411.30001]

Sabbah, Claude, Riemann-Hilbert correspondence, irregular singularities and Hodge theory, 315-326 [Zbl 1405.14026]

Schlichenmaier, Martin, Krichever-Novikov type algebras and Wess-Zumino-Novikov-Witten models, 327-368 [Zbl 1461.17026]

Schumacher, Georg, Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds, 369-399 [Zbl 1411.32018]

van Straten, Duco, Calabi-Yau operators, 401-451 [Zbl 1405.14027]

Tseng, Hsian-Hua, A survey on toric mirror symmetry, 453-473 [Zbl 1404.14051]

Veech, William A., Dynamical systems on analytic manifolds of quadratic differentials: chapter I, F-structures, 475-525 [Zbl 1428.37034]

Wang, Chin-Lung, Aspects on Calabi-Yau moduli, 527-550 [Zbl 1411.32019]

Zhang, Yuguang, Degeneration of Ricci-flat Calabi-Yau manifolds and its applications, 551-592 [Zbl 1411.32020]

Zhou, Jie, Mirror symmetry for plane cubics revisited, 593-619 [Zbl 1405.14106]

##### MSC:

14-06 | Proceedings, conferences, collections, etc. pertaining to algebraic geometry |

11-06 | Proceedings, conferences, collections, etc. pertaining to number theory |

30-06 | Proceedings, conferences, collections, etc. pertaining to functions of a complex variable |

53-06 | Proceedings, conferences, collections, etc. pertaining to differential geometry |

81-06 | Proceedings, conferences, collections, etc. pertaining to quantum theory |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

30F10 | Compact Riemann surfaces and uniformization |

14J33 | Mirror symmetry (algebro-geometric aspects) |

53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |

81R15 | Operator algebra methods applied to problems in quantum theory |

00B25 | Proceedings of conferences of miscellaneous specific interest |

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\textit{L. Ji} (ed.) and \textit{S.-T. Yau} (ed.), Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations. Based on the conference, Institute Mittag-Leffler, Stockholm, Sweden, July 13--18, 2015. Somerville, MA: International Press; Beijing: Higher Education Press (2018; Zbl 1398.14003)