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Graphs and Patterns in Mathematics and Theoretical Physics
Edited by: Mikhail Lyubich, Stony Brook University, NY, and University of Toronto, ON, Canada, and Leon Takhtajan, Stony Brook University, NY

Proceedings of Symposia in Pure Mathematics
2005; 418 pp; hardcover
Volume: 73
ISBN-10: 0-8218-3666-8
ISBN-13: 978-0-8218-3666-8
List Price: US$103
Member Price: US$82.40
Order Code: PSPUM/73
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The Stony Brook Conference, "Graphs and Patterns in Mathematics and Theoretical Physics", was dedicated to Dennis Sullivan in honor of his sixtieth birthday. The event's scientific content, which was suggested by Sullivan, was largely based on mini-courses and survey lectures. The main idea was to help researchers and graduate students in mathematics and theoretical physics who encounter graphs in their research to overcome conceptual barriers.

The collection begins with Sullivan's paper, "Sigma models and string topology," which describes a background algebraic structure for the sigma model based on algebraic topology and transversality. Other contributions to the volume were organized into five sections: Feynman Diagrams, Algebraic Structures, Manifolds: Invariants and Mirror Symmetry, Combinatorial Aspects of Dynamics, and Physics. These sections, along with more research-oriented articles, contain the following surveys: "Feynman diagrams for pedestrians and mathematicians" by M. Polyak, "Notes on universal algebra" by A. Voronov, "Unimodal maps and hierarchical models" by M. Yampolsky, and "Quantum geometry in action: big bang and black holes" by A. Ashtekar.

This comprehensive volume is suitable for graduate students and research mathematicians interested in graph theory and its applications in mathematics and physics.


Graduate students and research mathematicians interested in graph theory and its applications in mathematics and physics.

Table of Contents

  • D. Sullivan -- Sigma models and string topology
Feynman diagrams
  • M. Polyak -- Feynman diagrams for pedestrians and mathematicians
  • D. Kreimer -- Structures in Feynman graphs: Hopf algebras and symmetries
Algebraic structures
  • A. A. Voronov -- Notes on universal algebra
  • D. Tamarkin and B. Tsygan -- The ring of differential operators on forms in noncommutative calculus
  • V. Gorbounov, F. Malikov, and V. Schechtman -- Twisted chiral de Rham algebras on \(\mathbb{P}^1\)
Manifolds: Invariants and mirror symmetry
  • R. Kashaev and N. Reshetikhin -- Invariants of tangles with flat connections in their complements
  • S. Garoufalidis and J. Levine -- Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism
  • K. Fukaya -- Multivalued Morse theory, asymptotic analysis and mirror symmetry
Combinatorial aspects of dynamics
  • R. Forman -- Some applications of combinatorial differential topology
  • A. de Carvalho -- Extensions, quotients and generalized pseudo-Anosov maps
  • M. Yampolsky -- Unimodal maps and hierarchical models
  • A. Ashtekar -- Quantum geometry in action: big bang and black holes
  • P. van Nieuwenhuizen -- Supersymmetry, supergravity, superspace and BRST symmetry in a simple model
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