xii INTRODUCTION resolved via a proper birational morphism ˇ ψ → Xψ/G with ˇ ψ a new Calabi-Yau threefold with Hodge numbers h1,1( ˇ ψ ) = 101, h1,2( ˇ ψ ) = 1. These examples were already a surprise to mathematicians, since at the time very few examples of Calabi-Yau threefolds with positive Euler characteristic were known (the Euler characteristic coinciding with 2(h1,1 − h1,2)). Much more spectacular were the results of Candelas, de la Ossa, Green and Parkes [10]. Guided by string theory and path integral calculations, Candelas et al. conjectured that certain period calculations on the family ˇ ψ parameterized by ψ would yield predictions for numbers of rational curves on the quintic threefold. They carried out these calculations, finding agreement with the known numbers of rational curves up to degree 3. We omit any details of these calculations here, as they have been exposited in many places, see e.g., [43]. This agreement was very surprising to the mathematical community, as these numbers become increasingly diﬃcult to compute as the degree increases. The number of lines, 2875, was known in the 19th century, the number of conics, 609250, was computed only in 1986 by Sheldon Katz [66], and the number of twisted cubics, 317206375, was only computed in 1990 by Ellingsrud and Strømme [22]. Throughout the history of mathematics, physics has been an important source of interesting problems and mathematical phenomena. Some of the interesting mathematics that arises from physics tends to be a one-off — an interesting and unexpected formula, say, which once verified mathematically loses interest. Other contributions from physics have led to powerful new structures and theories which continue to provide interesting and exciting new results. I like to believe that mirror symmetry is one of the latter types of subjects. The conjecture raised by Candelas et al., along with related work, led to the study of Gromov-Witten invariants (defining precisely what we mean by “the num- ber of rational curves”) and quantum cohomology, a way of deforming the usual cup product on cohomology using Gromov-Witten invariants. This remains an active field of research, and by 1996, the theory was suﬃciently developed to allow proofs of the mirror symmetry formula for the quintic by Givental [34], Lian, Liu and Yau [75] and subsequently others, with the proofs getting simpler over time. Concerning mirror symmetry, Batyrev [6] and Batyrev-Borisov [7] gave very general constructions of mirror pairs of Calabi-Yau manifolds occurring as complete intersections in toric varieties. In 1994, Maxim Kontsevich [68] made his funda- mental Homological Mirror Symmetry conjecture, a profound effort to explain the relationship between a Calabi-Yau manifold and its mirror in terms of category theory. In 1996, Strominger, Yau and Zaslow proposed a conjecture, [108], now referred to as the SYZ conjecture, suggesting a much more concrete geometric relationship between mirror pairs namely, mirror pairs should carry dual special Lagrangian fibrations. This suggested a very explicit relationship between a Calabi-Yau man- ifold and its mirror, and initial work in this direction by myself [37, 38, 39] and Wei-Dong Ruan [97, 98, 99] indicates the conjecture works at a topological level. However, to date, the analytic problems involved in proving a full-strength ver- sion of the SYZ conjecture remain insurmountable. Furthermore, while a proof of the SYZ conjecture would be of great interest, a proof alone will not explain the finer aspects of mirror symmetry. Nevertheless, the SYZ conjecture has motivated

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