國家理論科學研究中心(南區)數學組

Coordinator:

Eugene Xia (NCKU)

Algebraic Geometry

zxia@math.ncku.edu.tw

Tel: +886-6-2757575 Ext. 65159

 

Participants:

  1. Roger Chen (NCKU)
  2. Yungyen Chiang (NSYSU)
  3. Nan-Kuo Ho (NCKU)
  4. Ying-Ji Hong (NCKU)

 

Topics:

This project will be devoted to the following two topics

  1. Interactions between Differential Geometry, Partial Differential Equations, and Algebraic Geometry
    We shall study some interactions between differential geometry, partial differential equations, and algebraic geometry. More specifically, we expect to understand further these deep connections between Differential Geometry, Partial Differential Equations, Complex Algebraic Geometry, and Symplectic Geometry.
  2. Higgs Bundles, Moduli Spaces and Dynamical Systems
    We shall study some moduli spaces on Riemann surfaces. A moduli space is a geometric object, such as a manifold or variety that parameterizes a family of objects in a universal way. These spaces play a central role in many branches of mathematics. The moduli spaces under study in this proposal arise from algebraic curves. They are

    • Betti spaces: the moduli of semi-simple representations of the fundamental groups,
    • De Rham spaces: the moduli of the semi-simple flat connections,
    • Dolbeault spaces: the moduli of semi-stable Higgs bundles.

 

We propose to study the topologies, the geometries of these moduli spaces and the dynamics of the mapping class group action on the Betti moduli space. For topology and geometry, we wish to identify the irreducible components of the Dolbeault moduli space for various structure groups. The mapping class group of a Riemann surface plays an important role in the study of three manifolds and mathematical physics. This group acts naturally on the Betti moduli space which we propose to study the measure and topological dynamics of this mapping class group action on the Betti moduli space.