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The focus of this book is the continuing strength of pure mathematics in Russia after the post-Soviet diaspora. The authors are eight young specialists who are associated with strong research groups in Moscow and St. Petersburg in the fields of algebraic geometry and number theory. Their articles are based on lecture courses given at British universities. The articles are mainly surveys of the recent work of the research groups and contain a substantial number of original results. Topics covered are embeddings and projective duals of homogeneous spaces, formal groups, mirror duality, del Pezzo fibrations, Diophantine approximation and geometric quantization. The authors are I. Arzhantsev, M. Bondarko, V. Golyshev, M. Grinenko, N. Moshchevitin, E. Tevelev, D. Timashev and N. Tyurin. Mathematical researchers and graduate students in algebraic geometry and number theory worldwide will find this book of great interest.