Transversal Research Project 1 : Noncommutative manifolds and discrete geometric objects in the framework of noncommutative 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 Noncommutative Geometry (NCG).
In particular, a wide range of topics including the AtiyahSinger Index
Theory, KTheory, 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 noncommutation or quantization leads to
the notion of Noncommutative manifolds.
The objective of our project is the followings:
1) Number Theory
Lfunctions, Noncommutative algebraic geometry
2) Noncommutative 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) Microlocal Analysis and Integrable Systems
Hyperfunctions, Psuedodifferential calculus, Poisson geometry,
Quantum integrable systems
6) Theoretical Physics
String theory, Moduli spaces, Mirror symmetry, SeibergWitten theory


