Topics in Geometry (2012 Fall)

Room: 綜三631
Time: T3T4 W5
期末報告: Jan 16, 10am-no later than 3pm
報告主題: 除上課提的那兩個問題,也可以講 Wells's Appendix, or Mukai's Chapter 9 or Chapter 10, or Kobayashi's Chapter 7。若你對另外的主題有興趣也可以，但請先與我討論看是否與這門課相關。另外請最晚在報告前一星期讓我知道你要講的題目。

Course Description
We would like to introduce varies moduli spaces this semester. The material that we would like to cover in the course includes the following:

Basic background (Lie group & group actions)

Fiber bundles

Connections on Principal bundles : principal connections, flat connections, de Rham moduli space

Holonomy, representations, Betti moduli space

Connections on Vector bundles: complex vector bundle over real manifolds, complex vector bundle over complex manifolds, Hermitian moduli space, Holomorphic moduli space

From de Rham moduli space to Hermitian moduli space.

Symplectic geometry on the de Rham moduli space.

References (待續)
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:

R. Wells, Differential Analysis on Complex manifolds, third edition

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:

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

Syllabus

9/18: Lie group and group actions

9/19: Fiber bundles

9/25: Associate bundles

9/26: Gauge transformations

10/2: Connections

10/3: Parallel transport

10/9: Covariant derivative and Curvature

10/10: no class (national holiday)

10/16: Gauge actions on connections

10/17: Yang-Mills functional on the space of connections

10/23: Moduli of flat bundles

10/24: Stabilizer of gauge action and centralizer of holonomy

10/30: Holonomy and the surface group representation

10/31: Topology of surfaces

11/6: Classification of principal G-bundles

11/7: Complex vector bundles on real manifold

11/13: Complex vector bundles on complex manifold

11/14: 二十分鐘

11/20: no class (one hour on 11/28, another on 1/2)

11/21: no class (school holiday)

11/27: 報告

11/28: 補課一小時

12/4: Holomorphic vector bundles

12/5: Action on Holomorphic bundle structures

12/11: Hermitian vector bundles

12/12: Action on Hermitian connections

12/18: Relation between Principal connections and connections of Vector bundle

12/19: Relation between Principal connections and connections of Vector bundle

12/25*: Linear symplectic space

12/26*: Linear symplectomorphism

1/1: no class (national holiday)

1/2*: Symplectic manifold and Kahler manifold補課一小時

1/8*: Hamiltonian action and Symplectic reduction (2-d gauge example)

1/9*: Symplectic reduction (moduli space example)

1/16*: 期末報告 (Start at 10am)

Evaluation

Final presentation

Last Updated: Dec 24, 2012
URL: http://www.math.nthu.edu.tw/~nankuo/TG.html