Foundations of Mathematics and Physics One Century After Hilbert

1. Front Matter

   Pages i-xxi

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2. Where We Stand Today

   • Joseph Kouneiher

   Pages 1-73

3. Mie’s Electromagnetic Theory of Matter and the Background to Hilbert’s Unified Foundations of Physics

   • Leo Corry

   Pages 75-96

4. Hilbert and Einstein

   • Joseph Kouneiher, John Stachel

   Pages 97-106

5. Grothendieck’s Unifying Vision of Geometry

   • Colin McLarty

   Pages 107-127

6. Understanding the 6-Dimensional Sphere

   • Michael Atiyah

   Pages 129-133

7. A Dozen Problems, Questions and Conjectures About Positive Scalar Curvature

   • Misha Gromov

   Pages 135-158

8. Geometry and the Quantum

   • Alain Connes

   Pages 159-196

9. What Every Physicist Should Know About String Theory

   • Edward Witten

   Pages 197-210

10. Quanta of Space-Time and Axiomatization of Physics

    • Ali H. Chamseddine

    Pages 211-251

11. Twistor Theory as an Approach to Fundamental Physics

    • Roger Penrose

    Pages 253-285

12. What Are We Missing in Our Search for Quantum Gravity?

    • Lee Smolin

    Pages 287-304

13. A Schema for Duality, Illustrated by Bosonization

    • Sebastian De Haro, Jeremy Butterfield

    Pages 305-376

14. The Dressing Field Method of Gauge Symmetry Reduction, a Review with Examples

    • J. Attard, J. François, S. Lazzarini, T. Masson

    Pages 377-415

15. Syntactic Phylogenetic Trees

    • Kevin Shu, Sharjeel Aziz, Vy-Luan Huynh, David Warrick, Matilde Marcolli

    Pages 417-441

This book explores the rich and deep interplay between mathematics and physics one century after David Hilbert’s works from 1891 to 1933, published by Springer in six volumes. The most prominent scientists in various domains of these disciplines contribute to this volume providing insight to their works, and analyzing the impact of the breakthrough and the perspectives of their own contributions. The result is a broad journey through the most recent developments in mathematical physics, such as string theory, quantum gravity, noncommutative geometry, twistor theory, Gauge and Quantum fields theories, just to mention a few. The reader, accompanied on this journey by some of the fathers of these theories, explores some far reaching interfaces where mathematics and theoretical physics interact profoundly and gets a broad and deep understanding of subjects which are at the core of recent developments in mathematical physics. The journey is not confined to the present state of the art, butsheds light on future developments of the field, highlighting a list of open problems. Graduate students and researchers working in physics, mathematics and mathematical physics will find this journey extremely fascinating.  All those who want to  benefit from a comprehensive description of all the latest advances in mathematics and mathematical physics, will find this book very useful too.

• Minkowski space
• Noncommutative geometry
• K-homology
• Relationship between physics and mathematics
• K-theory
• Heisenberg commutation relations
• Twistor theory
• Bosonization
• Pati-Salam symmetries
• Hilbert’s sixth problem
• quantum groups
• Virasoro algebra
• Penrose's quasi-local mass construction
• Thirring model
• massive integrable field theory
• string theory
• quantum gravity
• gauge theory
• Hilbert's foundations of physics
• Riemannian geometry

“This wonderful book is full of such insights, reflections, analyses, appraisals, and even anecdotes, and should be a smash hit: we should all seek to know more about these foundations, and we have a wonderful set of essays here, based off Hilbert’s original wonderful lectures, and a lot more.” (Michael Berg, MAA Reviews, July, 2018)

• Nice and Sophia Antipolis University, Nice, France

  Joseph Kouneiher

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