Transversal Research Project 1 : Noncommutative manifolds and discrete geometric objects in the framework of non-commutative geometry

Leader: Yoshiaki Maeda Sub Leader: Hitoshi Moriyoshi Aim of the project In this project we focus on making advances in the following areas in Mathematics from the viewpoint of Non-commutative Geometry (NCG). In particular, a wide range of topics including the Atiyah-Singer Index Theory, K-Theory, Combinatorics, Graph Theory, Erogodic Theory, Number Theory, Particle Physics and Integrable Systems have deep connections with NCG. In recent work, a close relationship between NCG and Particle Physics, String Theory and Geometry of Riemann Surfaces has been established. These investigations have also led to the study of Integrable Systems, Quantum Cohomology and Microlocal Analysis from a NCG point of view. The general process of non-commutation or quantization leads to the notion of Non-commutative manifolds. The objective of our project is the followings: 1) Number Theory L-functions, Non-commutative algebraic geometry 2) Non-commutative Geometry and Topology Geometric Quantization, Deformation Quantization of Poisson Geometry, Index Theory, Gauge Theory, String Theory, Quantum Field Theory 3) Discrete Mathematics Discrete Geometry, Graph Theory, Combinatorics Geometric quantization of lattices 4) Dynamical Systems Ergodic Theory, Integrable Systems, Dynamics and Number Theory, 5) Micro-local Analysis and Integrable Systems Hyperfunctions, Psuedo-differential calculus, Poisson geometry, Quantum integrable systems 6) Theoretical Physics String theory, Moduli spaces, Mirror symmetry, Seiberg-Witten theory