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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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Publisher’s description: The uniformization theorem of Riemann surfaces is one of the most beautiful and important theorems in mathematics. Besides giving a clean classification of Riemann surfaces, its proof has motivated many new methods, such as the Riemann-Hilbert correspondence, Picard-Fuchs equations, and higher-dimensional generalizations of the uniformization theorem, which include Calabi-Yau manifolds.
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]
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