We would like to understand varies moduli spaces this semester. Basic knowledge on differential geometry is required. Possible topics are:
Connections on real vector bundles: parallel transport, flat connections, Yang-Mills functional
Connections on Principal bundles : principal connections, flat connections, moduli space of flat connections
Holonomy, representations, moduli space of surface group representations
Connections on Vector bundles: complex vector bundle over real manifolds, complex vector bundle over complex manifolds, Hermitian moduli space, Holomorphic moduli space
Stability of holomorphic vector bundles, moduli space of stable vector bundles
Higgs bundles
Meromorphic connections
Geometry and Topology on the moduli space.
References («ÝÄò)
Background:
L. Duistermaat, Lie groups (1.1-1.4: Lie group and Lie algebra; 2.1-2.4: group actions and associate bundles)
J. Milnor, Characteristic classes (1-3: manifolds & vector bundles)
Motivation:
M. Atiyah, The Geometry and Physics of Knots
Physics side:
V. Guillemin and S. Sternberg, Symplectic techniques in Physics (III. Motion in a Yang-Mills field)
Principal connections and Holonomy:
S. Kobayashi, Foundations of Differential Geometry I & II
Vector bundles and their moduli:
J. Jost, Riemannian Geometry and Geometric analyis (3, 4.1-4.2)
S. Kobayashi, Differential Geometry of complex vector bundles
K. Donaldson, The Geometry of four manifolds, ( Chapter 2,4,6)
R. Gunning, Lectures on vector bundles over Riemann surfaces
S. Mukai, An introduction to invariants and moduli
Complex geometry:
R. Wells, Differential Analysis on Complex manifolds, third edition
D. Huybrechts, Complex Geometry
Related papers:
Atiyah and Bott, The Yang-Mills equations over Riemann surfaces
M. Atiyah, Collected works, Volume 5 - Gauge theories
N. Hitchin, Self-duality equations on a Riemann surface
P. Boalch, Quasi-Hamiltonian geometry of meromorphic connections
Syllabus
2/25: Vector bundle over real manifold
3/4
3/11
3/18
3/25
4/1
4/8
4/15
4/22
4/29
5/6
5/13
5/20
5/27
6/3
6/10
6/17
6/24
Evaluation
In-class presentation
Last Updated: Feb 25, 2015
URL: http://www.math.nthu.edu.tw/~nankuo/TM.html