In recent decades, quantization has led to interesting applications in various mathematical branches. This volume, comprised of research and survey articles, discusses key topics, including symplectic and algebraic geometry, representation theory, quantum groups, the geometric Langlands program, quantum ergodicity, and non-commutative geometry. A wide range of topics related to quantization are covered, giving a glimpse of the broad subject. The articles are written by distinguished mathematicians in the field and reflect subsequent developments following the Arithmetic and Geometry around Quantization conference held in Istanbul. List of Contributors: S. Akbulut, R. Hadani, S. Arkhipov, K. Kremnizer, Ö. Ceyhan, S. Mahanta, E. Frenkel, S. Salur, K. Fukaya, G. Ben Simon, D. Gaitsgory, W. van Suijlekom, S. Gurevich. Mirror Duality Via G2 And Spin (7) Manifolds / Selman Akbulut And Sema Salur -- 2-gerbes And 2-tate Spaces / Sergey Arkhipov And Kobi Kremnizer -- The Geometry Of Partial Order On Contact Transformations Of Pre-quantization Manifolds / Gabi Ben Simon -- Towards Quantum Cohomology Of Real Varieties / Özgür Ceyhan -- Weyl Modules And Opers Without Monodromy / Edward Frenkel And Dennis Gaitsgory -- Differentiable Operads, The Kuranishi Correspondence, And Foundations Of Topological Field Theories Based On Pseudo-holomorphic Curves / Kenji Fukaya -- Notes On The Self-reducibility Of The Weil Representation And Higher-dimensional Quantum Chaos / Shamgar Gurevich And Ronny Hadani -- Notes On Canonical Quantization Of Symplectic Vector Spaces Over Finite Fields / Shamgar Gurevich And Ronny Hadani -- Noncommutative Geometry In The Framework Of Differential Graded Categories / Snigdhayan Mahanta -- Multiplicative Renormalization And Hopf Algebras / Walter D. Van Suijlekom. Ozgur Ceyhan, Yuri I. Manin, Matilde Marcolli, Editors. Includes Bibliographical References. Front Matter....Pages i-viii Mirror Duality via G 2 and Spin (7) Manifolds....Pages 1-21 2-Gerbes and 2-Tate Spaces....Pages 23-35 The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds....Pages 37-64 Towards Quantum Cohomology of Real Varieties....Pages 65-99 Weyl Modules and Opers without Monodromy....Pages 101-121 Differentiable Operads, the Kuranishi Correspondence, and Foundations of Topological Field Theories Based on Pseudo-Holomorphic Curves....Pages 123-200 Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos....Pages 201-232 Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields....Pages 233-251 Noncommutative Geometry in the Framework of Differential Graded Categories....Pages 253-275 Multiplicative Renormalization and Hopf Algebras....Pages 277-292 This volume comprises both research and survey articles originating from the conference on Arithmetic and Geometry around Quantization held in Istanbul in 2006. A wide range of topics related to quantization are covered, thus aiming to give a glimpse of a broad subject in very different perspectives.
