Muutke küpsiste eelistusi

Gentle Introduction to Homological Mirror Symmetry [Pehme köide]

(Universiteit van Amsterdam)
  • Formaat: Paperback / softback, 400 pages, kõrgus x laius x paksus: 228x150x25 mm, kaal: 628 g, Worked examples or Exercises
  • Sari: London Mathematical Society Student Texts
  • Ilmumisaeg: 19-Aug-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108728758
  • ISBN-13: 9781108728751
Teised raamatud teemal:
Gentle Introduction to Homological Mirror SymmetrySuurem pilt
  • Formaat: Paperback / softback, 400 pages, kõrgus x laius x paksus: 228x150x25 mm, kaal: 628 g, Worked examples or Exercises
  • Sari: London Mathematical Society Student Texts
  • Ilmumisaeg: 19-Aug-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108728758
  • ISBN-13: 9781108728751
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781108728751.html
Märksõnad:
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.

Arvustused

'Each chapter concludes with a few exercises, and great care has been taken to use notation and terminology in a consistent way. That must have taken considerable effort and it greatly increases the value of the book because there are many examples in mirror symmetry of different writers using the same words to mean different, often subtly different, things.' G. K. Sankaran, MathSciNet 'The book under review provides an introduction to homological mirror symmetry which is accessible to graduate students in mathematics. In particular, it includes a great amount of background material, and motivational sections. Among the reason it is so approachable is the style ' Hulya Arguz, zbMATH

Muu info

Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.
Preface ix
PART ONE TO A∞ AND BEYOND
1(102)
1 Categories
3(12)
1.1 Categories
3(1)
1.2 Functors
4(2)
1.3 Natural Transformations
6(1)
1.4 Linear Categories
7(1)
1.5 Modules
8(2)
1.6 Morita Equivalence
10(2)
1.7 Exercises
12(3)
2 Cohomology
15(21)
2.1 Complexes
15(4)
2.2 Cohomology in Topology
19(8)
2.3 Cohomology in Algebra
27(7)
2.4 Exercises
34(2)
3 Higher Products
36(34)
3.1 Motivation and Definition
36(7)
3.2 Minimal Models
43(6)
3.3 A∞-Categories
49(12)
3.4 Bells and Whistles
61(8)
3.5 Exercises
69(1)
4 Quivers
70(33)
4.1 Representations of Quivers
70(4)
4.2 Strings and Bands
74(13)
4.3 Points and Sheaves
87(8)
4.4 Picturing the Categories
95(3)
4.5 A First Glimpse of Homological Mirror Symmetry
98(2)
4.6 Exercises
100(3)
PART TWO A GLANCE THROUGH THE MIRROR
103(146)
5 Motivation from Physics
105(19)
5.1 The Path Integral Formalism
105(2)
5.2 Symmetry
107(3)
5.3 Superstrings
110(3)
5.4 Categorical Interpretations
113(6)
5.5 What Is Mirror Symmetry?
119(2)
5.6 Exercises
121(3)
6 The A-Side
124(51)
6.1 Morse Theory
125(8)
6.2 The Basic Fukaya Category
133(12)
6.3 Variations
145(18)
6.4 Generators
163(9)
6.5 Exercises
172(3)
7 The B-Side
175(41)
7.1 Varieties
175(15)
7.2 Other Geometrical Objects
190(9)
7.3 Equivalences
199(13)
7.4 Exercises
212(4)
8 Mirror Symmetry
216(33)
8.1 The Complex Torus
216(2)
8.2 Toric Varieties
218(6)
8.3 Tropical Geometry
224(3)
8.4 One, Two, Three, Mirror Symmetry
227(7)
8.5 Away from the Large Limit
234(7)
8.6 Mirrors and Fibrations
241(4)
8.7 Exercises
245(4)
PART THREE REFLECTIONS ON SURFACES
249(124)
9 Gluing
251(44)
9.1 Marked Surfaces
252(7)
9.2 Gluing Arcs to Strings and Bands
259(10)
9.3 Gluing Fukaya Categories over Graphs
269(6)
9.4 Gluing and Mirror Symmetry
275(7)
9.5 Covers
282(3)
9.6 Dimer Models
285(7)
9.7 Mirrors Galore
292(1)
9.8 Exercises
293(2)
10 Grading
295(17)
10.1 Graded Surfaces
295(5)
10.2 Strings and Bands
300(1)
10.3 Characterizing Graded Surfaces
301(2)
10.4 Gradings and Matrix Factorizations
303(3)
10.5 Mirror Varieties
306(3)
10.6 Mirror Orbifolds
309(1)
10.7 Exercises
310(2)
11 Stabilizing
312(27)
11.1 The Grothendieck Group
312(3)
11.2 King Stability and Pair of Pants Decompositions
315(6)
11.3 Bridgeland Stability
321(6)
11.4 Stability Conditions and Quadratic Differentials
327(6)
11.5 Stability Manifolds
333(3)
11.6 Exercises
336(3)
12 Deforming
339(34)
12.1 Deformation Theory for/A∞-Algebras
339(8)
12.2 A∞-Extensions
347(1)
12.3 Deformation Theory for Gentle Algebras
348(3)
12.4 Filling the Pair of Pants
351(11)
12.5 Koszul Duality
362(3)
12.6 The Mirror Functor
365(4)
12.7 Exercises
369(4)
References 373(14)
Index 387
Raf Bocklandt is Lecturer in Mathematics at the University of Amsterdam.