A Gentle Introduction to Homological Mirror Symmetry (London Mathematical Society Student Texts, Series Number 99)

Homological mirror symmetry has its origins in theoretical physics but is now of great interest in mathematics due to the deep connections it reveals between different areas of geometry and algebra. This book offers a self-contained and accessible introduction to the subject via the representation theory of algebras and quivers. It is suitable for graduate students and others without a great deal of background in homological algebra and modern geometry. Each part offers a different perspective on homological mirror symmetry. Part I introduces the A-infinity formalism and offers a glimpse of mirror symmetry using representations of quivers. Part II discusses various A- and B-models in mirror symmetry and their connections through toric and tropical geometry. Part III deals with mirror symmetry for Riemann surfaces. The main mathematical ideas are illustrated by means of simple examples coming mainly from the theory of surfaces, helping the reader connect theory with intuition "This book grew out of an advanced masters course that I teach biannually at the University of Amsterdam. The course is aimed at students who are doing a masters in algebra and geometry or mathematical physics. In this course I try to give them the feeling of what homological mirror symmetry is and how it ties together many different areas of mathematics. The focus of the course is to explain the main concepts and results and to illustrate them with examples, without getting too technical. In this way the students will be better prepared to delve into the primary literature if they want to understand the theory at a deeper and more detailed level"-- Provided by publisher "This book grew out of an advanced masters course that I teach biannually at the University of Amsterdam. The course is aimed at students who are doing a masters in algebra and geometry or mathematical physics. In this course I try to give them the feeling of what homological mirror symmetry is and how it ties together many different areas of mathematics. The focus of the course is to explain the main concepts and results and to illustrate them with examples, without getting too technical. In this way the students will be better prepared to delve into the primary literature if they want to understand the theory at a deeper and more detailed level"--Résumé de l'éditeur 01.0_pp_i_iv_Frontmatter 02.0_pp_v_viii_Contents 03.0_pp_ix_xii_Preface 04.0_pp_1_2_To_A_And_Beyond 04.1_pp_3_14_Categories 04.2_pp_15_35_Cohomology 04.3_pp_36_69_Higher_Products 04.4_pp_70_102_Quivers 05.0_pp_103_104_Glance_Through_The_Mirror 05.1_pp_105_123_Motivation_from_Physics 05.2_pp_124_174_The_A-Side 05.3_pp_175_215_The_B-Side 05.4_pp_216_248_Mirror_Symmetry 06.0_pp_249_250_Reflections_On_Surfaces 06.1_pp_251_294_Gluing 06.2_pp_295_311_Grading 06.3_pp_312_338_Stabilizing 06.4_pp_339_372_Deforming 07.0_pp_373_386_References 08.0_pp_387_390_Index