Advanced Topics in Geometry and Topology (MAST90124)
--Vector bundles over Riemann surfaces.
Instructor Yaping Yang
Office: Peter Hall building, Office 165
Office Hours: Mondays 10-11:30pm, Thursdays 2-3:30 pm.
Email: yaping.yang1@unimelb.edu.au
Class meeting
Location:
Peter Hall-G03 (Evan Williams Theatre)
Time:
Monday 15:15-16:15. Thursday 11:00-12:00 Friday 11:00-12:00
Prerequisites(from the handbook):
Students must have completed one of the following subjects:
References:
1. [Hitchin]: Integrable Systems, Twistors, Loop Groups, and Riemann Surfaces, N. J. Hitchin, G. B. Segal, and R. S. Ward.
(We will focus on the lecture: Riemann Surfaces and integrable Systems by Hitchin, Page11--52.)
2.[Gunning 1966]: Lectures on Riemann surfaces, by R.C. Gunning, Princeton University Press 1966. (1966)
3.[Gunning 1967]: Lectures on vector bundles over Riemann surfaces, by R.C. Gunning, Princeton University Press 1967. (1967)
4.[Forster]: Lectures on Riemann Surfaces, by Otto Forster, Graduate Texts in Mathematics.
Background reading:
1. [Lee]: Introduction to smooth manifolds, by John Lee.
http://webmath2.unito.it/paginepersonali/sergio.console/lee.pdf
Assessment:
Up to 50 pages of written assignments over the course of the semester, worth a total of 50% of the mark. A 3-hour written examination during the examination period worth 50% of the mark.
Assignment 1: is due on Aug 20, 2018.
Assignment 2: is due on Sep 7, 2018 (extended to Sep 14, 2018)
Assignment 3: due on Oct 12, 2018.
!!! Final Review !!!
Pre Exam Consultation: Monday Oct. 29, 2018: 10am-1pm.
Tentative Schedule and lecture notes:
Week 1
Topics: Smooth manifolds, Riemann surfaces, holomorphic maps, examples.
23 July 2018:
Lecture notes 1, Read Page 1-4 of [Hitchin], Slides of Soibelman on "Moduli spaces of Higgs bundles in mathematics andphysics"
26 July 2018:
Lecture notes 2, Read Page 11-12 of [Hitchin], and read Section 1 of [Forster]
27 July 2018:
Lecture notes 3, Read Page 13-14 of [Hitchin], and read Section 2 of [Forster]
Week 2
Line bundles, the canonical bundle
30 July 2018:
Lecture notes 4, Read Page 14-17 of [Hitchin].
2 August 2018
Lecture notes 4.5. Read Page 219 of [Forster].
3 August 2018
See Lecture notes 4. Read Page 14-17 of [Hitchin].
Week 3
Sheaves, sheaf cohomology
6 August 2018
Lecture notes 5.5, Lecture notes 5.
Read [Lee] P41-52, P57-58 on tangent bundles, and P65-P70 on cotangent bundles.
Read [Hitchin] P17-P18.
9 August 2018
10 August 2018
Read [Hitchin] P19-21
Week 4
Properties of sheaf cohomology, the Picard group.
13 August 2018
16 August 2018
17 August 2018
Week 5
Serre Duality, De Rham cohomology, the De Rham theorem.
20 August 2018
23 August 2018
24 August 2018
Week 6
Vector bundles, rank and degree, sections, construction of new vector bundles, Chern classes, the Riemann–Roch theorem.
Classification of vector bundles on P^1
27 August 2018
30 August 2018
31 August 2018
Week 7
Direct image of line bundles, the Riemann Hurwitz genus formula.
Hyperelliptic Riemann surfaces.
3 September 2018
6 September 2018
7 September 2018
Week 8
The theta divisor, the Torelli theorem.
The Jacobian, Matrix polynomials and Lax pairs
10 September 2018
13 September 2018
14 September 2018
Week 9
The Jacobian, Matrix polynomials and Lax pairs
Symplectic manifolds, Hamiltonian vector fields.
17 September 2018
20 September 2018
21 September 2018
Monday 24 September to Sunday 30 September:
Non Teaching Period & UA Common Vacation Week
Week 10
Coadjoint orbits
Completely integrable hamiltonian systems
Calogero-Moser system.
1 October 2018
4 October 2018
5 October 2018
Week 11
(Semi)-stable vector bundles.
The Hitchin system: the moduli space of semi-stable rank r and degree d Higgs bundles
8 October 2018
11 October 2018
12 October 2018
Week 12
The Hitchin fibration
15 October 2018
18 October 2018
19 October 2018
Try to read Hitchin's paper.
Final:
2 November 2018, Friday.
8:30am-11:30am
HOW TO LEARN abstract MATHEMATICS (by Ivan Mirković, from http://people.math.umass.edu/~mirkovic/B.UndergraduateCourses/370/1.html)