The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning (Lecture Notes in Mathematics)

✍ Scribed by Yang-Hui He

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 214
• Category: Library

✦ Synopsis

Can artificial intelligence learn mathematics? The question is at the heart of this original monograph bringing together theoretical physics, modern geometry, and data science.

The study of Calabi–Yau manifolds lies at an exciting intersection between physics and mathematics. Recently, there has been much activity in applying machine learning to solve otherwise intractable problems, to conjecture new formulae, or to understand the underlying structure of mathematics. In this book, insights from string and quantum field theory are combined with powerful techniques from complex and algebraic geometry, then translated into algorithms with the ultimate aim of deriving new information about Calabi–Yau manifolds. While the motivation comes from mathematical physics, the techniques are purely mathematical and the theme is that of explicit calculations. The reader is guided through the theory and provided with explicit computer code in standard software such as SageMath, Python and Mathematica to gain hands-on experience in applications of artificial intelligence to geometry.

Driven by data and written in an informal style, The Calabi–Yau Landscape makes cutting-edge topics in mathematical physics, geometry and machine learning readily accessible to graduate students and beyond. The overriding ambition is to introduce some modern mathematics to the physicist, some modern physics to the mathematician, and machine learning to both.

✦ Table of Contents

Preface
Acknowledgements
Contents
1 Prologus Terræ Sanctæ
1.1 A Geometrical Tradition
1.1.1 A Modern Breakthrough
1.1.2 Preliminary Examples: 1,2,? …
1.2 10 = 4 + 2 3: A Physical Motivation
1.2.1 Triadophilia
1.2.2 Caveat and Apology for the Title
1.3 Topological Rudiments
1.3.1 The Hodge Diamond
2 The Compact Landscape
2.1 The Quintic
2.1.1 Topological Quantities: Exact Sequences
2.1.2 Topological Quantities: Computer Algebra
2.2 CICY: Complete Intersection Calabi–Yau
2.2.1 Topological Quantities: Statistics
2.3 Other Datasets
2.3.1 Hypersurfaces in Weighted CP4
2.3.2 Elliptic Fibrations
2.4 An Explosion: The Kreuzer–Skarke Dataset
2.4.1 Reflexive Polytopes
2.4.2 CY Hypersurfaces: Gradus ad Parnassum
2.4.3 1, 16, 4319, 473800776 …
2.5 Cartography of the Compact Landscape
2.6 Epilogue: Recent Developments
2.7 Post Scriptum: Calabi–Yau Metrics
2.7.1 Numerical Metric on the Quintic
3 The Non-Compact Landscape
3.1 Another 10 = 4 + 2 3
3.1.1 Quiver Representations and a Geometer's AdS/CFT
3.1.2 The Archetypal Example
3.2 Orbifolds and Quotient Singularities
3.2.1 McKay Correspondence
McKay Quiver for Z/(2 Z)
3.2.2 Beyond ADE
3.3 Toric Calabi-Yau Varieties
3.3.1 Cone over P2
3.3.2 The Conifold
3.3.3 Bipartite Graphs and Brane Tilings
3.4 Cartography of the Affine Landscape
3.4.1 Gorenstein Singularities
3.4.2 The Non-Compact Landscape
4 Machine-Learning the Landscape
4.1 A Typical Problem
4.1.1 WWJD
4.2 Rudiments of Machine-Learning
4.2.1 Regression and a Single Neuron
4.2.2 MLP: Forward Feeding Neural Networks
4.2.3 Convolutional Neural Networks
4.2.4 Some Terminology
4.2.5 Other Common Supervised ML Algorithms
Support Vector Machines
Decision Trees
Naïve Bayes Classifier
Nearest Neighbours
4.2.6 Goodness of Fit
4.3 Machine-Learning Algebraic Geometry
4.3.1 Warm-up: Hypersurface in Weighted P4
ML WP4-Hypersurfaces: Python
ML WP4-Hypersurfaces: Mathematica
Non-NN ML methods
4.3.2 Learning CICYs
4.3.3 More Success Stories in Geometry
4.4 Beyond Algebraic Geometry
4.5 Epilogue
5 Postscriptum
A Some Rudiments of Complex Geometry
A.1 (Co-)Homology
A.2 From Real to Complex to Kähler
A.3 Bundles and Sequences
A.4 Chern Classes
A.5 Covariantly Constant Spinor
A.6 A Lightning Refresher on Toric Varieties
A.6.1 Digressions on Spec and All That
A.6.2 Example: The Conifold
A.6.3 From Cones to Fans
A.7 Dramatis Personae
A.8 The Kodaira Classification of Elliptic Fibrations
B Gröbner Bases: The Heart of Computational Algebraic Geometry
B.1 An Elimination Problem
B.2 Hilbert Series
C Brane Tilings
C.1 Dessins d'Enfants
D Remembering Logistic Regression
E A Computational Compendium: Homage to SageMath
E.1 Algebraic Varieties
E.2 Combinatorics and Toric Varieties
E.3 Representation Theory
E.4 Number Theory and More
Glossary
References
Index