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The Calabi–Yau Landscape

From Geometry, to Physics, to Machine Learning

  • Book
  • © 2021

Overview

Authors:
  1. Yang-Hui He

    1. London Institute, Royal Institution of Great Britain; City, University of London; School of Physics & Chern Institute, Nankai University; Merton College, University of Oxford, Oxford, UK

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  • The first monograph applying machine learning to problems of geometry
  • Provides a data-driven introduction to computational algebraic geometry
  • Delivers a quick introduction to modern data science, with code in popular software (Python, SageMath and Mathematica)
  • Includes background in geometry, algebra, and theoretical physics

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2293)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xvii
    Download chapter PDF

  2. Prologus Terræ Sanctæ

    • Yang-Hui He
    Pages 1-18

  3. The Compact Landscape

    • Yang-Hui He
    Pages 19-58

  4. The Non-Compact Landscape

    • Yang-Hui He
    Pages 59-85

  5. Machine-Learning the Landscape

    • Yang-Hui He
    Pages 87-130

  6. Postscriptum

    • Yang-Hui He
    Pages 131-131

  7. Back Matter

    Pages 133-206
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Keywords

About this book

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.

Reviews

“The description of the geometry of Calabi–Yau manifolds at the beginning of the book is directed to readers with a solid geometric background … . The book mainly consists of a clear presentation on how neutral networks and machine learning tools can be usefully employed to interact (with a statistically high rate of success) with deep theoretical studies of geometric objects.” (Luca Chiantini, zbMATH 1492.14001, 2022)

“The book assists the reader in such outsourced learning. It is structured as a sort of hypertext that provides general context and a big picture with embedded ‘links’ (references) to the more specialized literature. … The message of the Calabi-Yau landscape is that it is time for both to learn data science, and this book is a fine place to start.” (Sergiy Koshkin, Mathematical Reviews, October, 2022)

Authors and Affiliations

  • London Institute, Royal Institution of Great Britain; City, University of London; School of Physics & Chern Institute, Nankai University; Merton College, University of Oxford, Oxford, UK

    Yang-Hui He

About the author

Professor Yang-Hui He is a mathematical physicist working at the interface of geometry, number theory and quantum field theory/string theory. Recently, he helped introduce machine learning into the field of pure mathematics by using AI to help uncover new patterns and raise new conjectures (cf. interview by Science [Vol 365, July, 2019] and by New Scientist [Dec 9 Issue, 2019]). He has over 150 papers and 2 books, with more than 6500 citations, h-index 45 (Google Scholar). Professor He received his BA from Princeton University (summa cum laude), MA from Cambridge (distinction, Tripos) and PhD from MIT. He is currently Fellow of the London Institute, Royal Institution, jointly tutor in mathematics at Merton College, University of Oxford, professor of mathematics at City, University of London, and chair professor of physics at Nankai University.

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