Mikhail Lyubich

Leon Armenovich Takhtadzhi͡an

Dennis Sullivan

Conference on Graphs and Patterns in Mathematics and Theoretical Physics (2001 Stony Bro

Conference on Graphs and Patterns in Mathematics and Theoretical Physics

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.

Proceedings of the Conference on Graphs and Patterns in Mathematics and Theoretical Physics, Dedicated to Dennis Sullivan's 60th Birthday, June 14-21, 2001, Stony Brook University, Stony Brook, NY

Graphs and Patterns in Mathematics and Theoretical Physics

This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.

Quantum Mechanics for Mathematicians

Lee-Peng Teo

In this memoir, we prove that the universal Teichmuller space $T(1)$ carries a new structure of a complex Hilbert manifold and show that the connected component of the identity of $T(1)$ - the Hilbert submanifold $T_{0}(1)$ - is a topological group. We define a Weil-Petersson metric on $T(1)$ by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that $T(1)$ is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmuller curve fibration over the universal Teichmuller space.As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller space. We study in detail the Hilbert manifold structure on $T_{0}(1)$ and characterize points on $T_{0}(1)$ in terms of Bers and pre-Bers embeddings by proving that the Grunsky operators $B_{1}$ and $B_{4}$, associated with the points in $T_{0}(1)$ via conformal welding, are Hilbert-Schmidt. We define a 'universal Liouville action' - a real-valued function ${\mathbf S}_{1}$ on $T_{0}(1)$, and prove that it is a Kahler potential of the Weil-Petersson metric on $T_{0}(1)$.We also prove that ${\mathbf S}_{1}$ is $-\tfrac{1}{12\pi}$ times the logarithm of the Fredholm determinant of associated quasi-circle, which generalizes classical results of Schiffer and Hawley. We define the universal period mapping $\hat{\mathcal{P}}: T(1)\rightarrow\mathcal{B}(\ell^{2})$ of $T(1)$ into the Banach space of bounded operators on the Hilbert space $\ell^{2}$, prove that $\hat{\mathcal{P}}$ is a holomorphic mapping of Banach manifolds, and show that $\hat{\mathcal{P}}$ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan.We prove that the restriction of $\hat{\mathcal{P}}$ to $T_{0}(1)$ is an inclusion of $T_{0}(1)$ into the Segal-Wilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group $S$ of symmetric homeomorphisms of $S^{1}$ under the mapping $\hat{\mathcal{P}}$ consists of compact operators on $\ell^{2}$. The results of this memoir were presented in our e-prints: Weil-Petersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004).

Leon Armenovich Takhtadzhi͡an

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